Statistical Inference and Hypothesis Testing in Data Science Applications Coursera Quiz Answers 2022 | All Weeks Assessment Answers [💯Correct Answer]

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Statistical Inference and Hypothesis Testing in Data Science Applications Coursera Quiz Answers 2022 | All Weeks Assessment Answers [💯Correct Answer]
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Hello Peers, Today we are going to share all week’s assessment and quizzes answers of the Statistical Inference and Hypothesis Testing in Data Science Applications course launched by Coursera totally free of cost✅✅✅. This is a certification course for every interested student.

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About Statistical Inference and Hypothesis Testing in Data Science Applications Course

This course will focus on the theory and practise of testing hypotheses, especially as they relate to data science. Students will learn how to use hypothesis tests so that they can use data to make good decisions. The general logic of testing hypotheses, error and error rates, power, simulation, and the right way to calculate and understand p-values will be given extra attention. Also, the wrong use of testing ideas, especially p-values, and the ethical consequences of that use will be looked at.

This course is part of CU Boulder’s Master of Science in Data Science (MS-DS) degree, which is offered on the Coursera platform. It can be taken for academic credit. The MS-DS is a degree that brings together professors from Applied Mathematics, Computer Science, Information Science, and other departments at CU Boulder. The MS-DS is perfect for people who have a wide range of undergraduate education and/or work experience in computer science, information science, mathematics, and statistics. Admission is based on performance, and there is no application process. Visit https://www.coursera.org/degrees/master-of-science-data-science-boulder to find out more about the MS-DS program.

WHAT YOU’LL FIND OUT

  • Explain what a composite hypothesis is and how important a test is when the null hypothesis is a composite one.
  • Explain what a test statistic, significance level, and rejection region are for a hypothesis test. Give the shape of a no-go zone.
  • Do tests to find out what the real population variance is.
  • Find the sample mean and sample a minimum of the exponential distribution’s sampling distributions.

Course Apply Link – Statistical Inference and Hypothesis Testing in Data Science Applications

Statistical Inference and Hypothesis Testing in Data Science Applications Quiz Answers

Week 1 Quiz Answers

Quiz 1: Introduction to Hypothesis Testing Quiz Answers

Q1. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40.

If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction.

To verify the amperage of the fuses, a sample of fuses is to be selected and inspected. A hypothesis test is performed on

H_{0}: \mu = 40 \,\,\, \text{versus} \,\,\, H_{1}: \mu > 40H

Which of the following statements are true for describing Type I and Type I errors. (Check all that apply.)

  • Type I: The mean amperage is good and the manufacturer concludes that it is acceptable, which saves the manufacturer money.
  • T​ype II: The mean amperage is greater than 40, but the manufacturer concludes that it is less than 40.
  • T​ype II: The mean amperage is too high, but the manufacturer concludes that it is acceptable which opens them up to liability for damaged electrical systems.
  • T​ype I: The mean amperage is good, but the manufacturer concludes that it is too high and they launch an expensive and unnecessary recall process.

Q2. Minor surgery on horses under field conditions requires a reliable short-term anesthetic producing good muscle relaxation, minimal cardiovascular and respiratory changes, and a quick, smooth recovery with minimal after effects so that horses can be left unattended.

A study reports that for a sample of n=73n=73 horses to which ketamine is administered under certain conditions, the sample average lateral recumbency (lying-down) time was 18.8618.86 min.

Lateral recumbency time is known to be normally distribited with a standard deviation of \sigma=8.6σ=8.6 min.

Do these data suggest that true average lateral recumbency time under these conditions is less than 20 min?

Test the hypotheses

​H_{0} :\mu = 20 \,\,\, \text{versus} \,\,\, H_{1}: \mu< 20​H
​
:μ=20versusH
​
:μ<20

at level of significance 0.100.10.

G​ive the appropriate critical value and conclusion.

  • 1​.28, Fail to reject H_{0}H
    ​
  • 1.28, Reject H_{0}H
    ​
  • -​1.28, Reject H_{0}H
    ​
  • -1.28, Fail to reject H_{0}H

Q3. The recommended daily dietary allowance for zinc among males older than age 50 years is 15 mg/day. A study reports the following summary data on intake for a sample of 12 males age 65-74 years: \overline{x} = 11.3

Assuming that the true distribution for zinc intake is normally distributed with variance 6.436.43, does the data indicate that the average daily zinc intake in the population of all males 65-74 falls below the recommended allowance?

C​onsider testing the hypotheses H_{0}: \mu=15H
​
:μ=15 versus H_{1}: \mu< 15H
​
:μ<15 at level of significance \alpha=0.05α=0.05.

B​elow what value does \overline{x}

need to be for us to reject H_{0}H
​
in favor of H_{1}H
​
? Give your answer to two decimal places.

Enter answer here

Q4. The director of manufacturing for ACME industries is interested in an online training program that can be used to train the firm’s maintenence employees for machine-repair operations.

  • To evaluate the training method, the director of manufacturing has agreed to train 15 employees with the new approach. The director will approve the program as long as the mean training time stays below 50 days. However, the director firmly believes that it will take, on average, 50 or more days.
  • Assuming that training times are normally distributed with a standard deviation of \sigma=3.2σ=3.2 days, what is the largest sample mean that could be observed that will make the company switch to the online program when testing the relevant hypotheses at level \alpha=0.01α=0.01?

Round your answer to 2 decimal places.

Enter answer here

Q5. W​hich of the following statements is necessarily true in hypothesis testing.

  • A​ Type I error is more serious/consequential than a Type II error.
  • A​ Type II error is more serious/consequential than a Type I error.
  • B​oth Type I and Type II errors are equally serious/consequential.
  • T​he relative severity of one error over the other is problem dependent.

Week 2 Quiz Answers

Quiz 1: Constructing Tests Quiz Answers

Q1. A​ commonly prescribed drug for relieving anxiety is believed to 60% effective. Experimental results with a new drug administered to a random sample of 100 adults who are suffering with anxiety showed that 68 experienced relief.

  • I​s this sufficient evidence to conclude that the new drug is superior to the one commonly prescribed?
  • Carry out the relevant hypothesis test using a 0.03 level of significance. Give the appropriate critical value and decision.
  • 1​.88, Reject H_{0}H
    0
    ​
    . The new drug appears to be superior at 0.03 level of significance.
  • -0.97, Fail to reject H_{0}H
    0
    ​
    . The new drug does not appear to be superior at 0.03 level of significance.
  • 1​.88, Fail to reject H_{0}H
    0
    ​
    . The new drug does not appear to be superior at 0.03 level of significance.
  • -1​.88, Reject H_{0}H
    0
    ​
    . The new drug appears to be superior at 0.03 level of significance.

Q2. “Hardness” of a material in engineering is defined as the resistance to indentation. It is determined by measuring the permanent depth of the indentation.

  • An engineer measured the “Brinell hardness” of 6 pieces of ductile iron
    that were subcritically annealed. The resulting data were:
  • 179, 156, 167,183, 178, 165
  • Assuming that Brinell hardness is normally distributed with standard deviation \sigma=4.2σ=4.2, consider testing
  • H_{0}: \mu = 170H
    0
    ​
    :μ=170 versus H_{1}: \mu>170H
    1
    ​
    :μ>170.

What is the P-Value for this test and what is the conclusion based on this data set at 0.20 level of significance?

  • 0​.8416, Fail to Reject H_{0}H
    0
    ​
    .
  • 0​.2184, Fail to Reject H_{0}H
    0
    ​
    .
  • 0​.7776, Fail to Reject H_{0}H
    0
    ​
    .
  • 0​.8416, Reject H_{0}H
    0

Q3. W​hich option below best describes the following two plots?

  • P​lot A indicates a heavy-tailed distribution and Plot B indicates a light-tailed distribution.
  • B​oth Plots A and B indicate heavy tailed distributions.
  • P​lot A indicates a light-tailed distribution and Plot B indicates a normal distribution.
  • P​lot A indicates a light-tailed distribution and Plot B indicates a heavy-tailed distribution.

Q4. A​ random sample of 10 chocolate energy bars from a certain company has, on average, 232 calories with a standard deviation of 15 calories.

W​hich of the following is a desirable shape of the power function for this test?

Q5. A manufacturer of ketchup uses a machine to automatically dispense
ingredients into bottles that move along an assembly line. The machine is working properly when 14 ounces are dispensed.

Suppose that the amount dispensed is normally distributed with a true standard deviation of 0.08 ounces. A random sample of 20 bottles had a sample mean of 13.8 ounces per bottle.

Is there evidence that the machine should be stopped and recalibrated? Test the relevant hypotheses at 0.05 level of significance.

W​hich of the following is true.

  • R​eject that the machine is properly calibrated if the sample mean is below 13.88 ounces or above 14.12 ounces.
  • R​eject that the machine is properly callibrated if the sample mean is below 13.76 ounces or above 14.84 ounces.
  • R​eject that the machine is properly calibrated if the sample mean is below 13.96 ounces or above 14.04 ounces.
  • R​eject that the machine is properly callibrated if the sample mean is below 13.6 ounces or above 14.4 ounces.

Week 2 Quiz Answers

Quiz 1: More Hypothesis Tests!

Q1. A manufacturer of 40-amp fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40.

If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the mean amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction.

To verify the amperage of the fuses, a sample of 10 fuses are selected and inspected. The average amperage at which fuses in the sample burned out was 38.8. The sample variance was 2.8.

A​ssuming burn out amperage is normally distributed, consider testing the hypotheses

H_{0}: \mu=40 \,\,\, \text{versus} \,\,\, H_{1}: \mu \ne 40H0​:μ=40versusH1​:μ​=40.

Find the P-value for this test.

  • 0​.0233
  • 0.0495
  • 0​.0248
  • 0.0467

Q2. The director of manufacturing for ACME industries is interested in a computer-assisted training program that can be used to train the firm’s maintenence employees for machine-repair operations. To evaluate the training method, the director of manufacturing has requested an estimate of the mean training time required with the computer assisted program. Suppose that management has agreed to train 15 employees with the new approach. The resulting sample mean training time is 53.87 days and the resulting sample standard deviation is 6.82 days.

Assuming that training times are normally distributed, is there evidence in the data to conclude that the true mean training time is greater than 50?

U​se \alpha=0.05α=0.05.

G​ive the appropriate test statistic, critical value, and conclusion.

  • T​he test statistic is approximately 2.22.2. The critical value is approximately 1.6451.645. The data does suggest that the true mean training time is greater than 50 when using \alpha=0.05α=0.05.
  • T​he test statistic is approximately 2.22.2. The critical value is approximately 1.761.76. The data does suggest that the true mean training time is greater than 50 when using \alpha=0.05α=0.05.
  • T​he test statistic is approximately 2.22.2. The critical value is approximately -1.76.−1.76. The data does not suggest that the true mean training time is greater than 50 when using \alpha=0.05α=0.05.
  • T​he test statistic is approximately 2.22.2. The critical value is approximately 1.701.70. The data does suggest that the true mean training time is greater than 50 when using \alpha=0.05α=0.05.

Q3. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample from the normal distribution with mean \muμ and variance \sigma^{2}σ2.

L​et S^{2}S2 be the sample variance.

W​hat is the distribution of S^{2}S2?

  • \Gamma \left( \frac{n-1}{2}, \frac{n-1}{2 \sigma^{2}} \right)Γ(2n−1​,2σ2n−1​)
  • \sigma^{2} \cdot \chi^{2}(2n)σ2⋅χ2(2n)
  • \Gamma \left( \frac{n-1}{2}, \frac{1}{2} \right)Γ(2n−1​,21​)
  • \chi^{2}(n-1)χ2(n−1)

Q4. A​ national supermarket chain wants to redesign self checkout lanes throughout the country. Two designs have been suggested. Data on customer checkout times, in minutes, have been collected at stores where the systems have already been installed. The results were as follows.

System ASystem B
s​ample size1​201​00
s​ample mean4​.13​.3
s​ample standard deviation2​.11​.5

Assuming normality of customer checkout times for both systems, consider testing

H_{0}: \mu_{A}=\mu_{B} \,\,\, \text{versus} \,\,\, H_{1}: \mu_{A}>\mu_{B},H0​:μA​=μB​versusH1​:μA​>μB​,

w​here \mu_{A}μA​ is the true mean for System A and \mu_{B}μB​ is the true mean for System B.

G​ive the value of the appropriate test statistic to 2 decimal places.

Q5. W​hen is Welch’s t-test used to compare two population means, and what distributional assumptions are needed?

  • Welch’s t-test is used when at least one of the two sample sizes are small, the true variances for the populations are unknown, and we can’t assume they are equal. W​e need to assume that the two populations are normally distributed.
  • Welch’s t-test is used when both sample sizes are small, the true variances for the populations are known. W​e need to assume that the two populations are normally distributed.
  • Welch’s t-test is used when both sample sizes are large and the true variances are unknown so we must use sample variances. We don’t need to assume normality because of the Central Limit Theorem.
  • Welch’s t-test is used when at least one of the two sample sizes are small, and the true variances for the populations are unknown but assumed to be equal. W​e need to assume that the two populations are normally distributed.

Week 4 Quiz Answers

Quiz 1: Best Tests and Some General Skills Quiz Answers

Q1. L​et X_{1}X1​ be a sample of size 1 from the continuous uniform distribution over the interval from 00 to \thetaθ. This means that X_{1}X1​ has pdf

​f(x) =

⎧⎩⎨1θ0,,0<x<θotherwise

​f(x)={θ1​0​,,​0<x<θotherwise​

C​onsider testing the hypotheses

​H_{0}: \theta=1 \,\,\, \text{versus} \,\,\, H_{1}: \theta<1​H0​:θ=1versusH1​:θ<1

a​t level of significance \alphaα, based on this single observation X_{1}X1​.

W​hich of the following is a valid test.

  • R​eject H_{0}H0​, in favor of H_{1}H1​ if X_{1} < \alphaX1​<α.
  • R​eject H_{0}H0​, in favor of H_{1}H1​ if X_{1} > \alphaX1​>α.
  • R​eject H_{0}H0​, in favor of H_{1}H1​ if X_{1} < 1- \alphaX1​<1−α.
  • R​eject H_{0}H0​, in favor of H_{1}H1​ if X_{1} >1- \alphaX1​>1−α.

Q2. Compute the following quantity.

​\chi^{2}_{0.01,9}+t_{0.90,6}​χ0.01,92​+t0.90,6​

G​ive your answer to two decimal places.

Q3. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample from the \Gamma(2,\beta)Γ(2,β) distribution.

Give the Neyman-Pearson ratio for finding the best test of

​H_{0}: \beta=2 \,\,\, \text{versus} \,\,\, H_{1}: \beta=3.H0​:β=2versusH1​:β=3.

  • \left( \frac{2}{3}\right)^{n} \displaystyle e^{\sum_{i=1}^{n} X_{i}}(32​)ne∑i=1n​Xi​
  • \left( \frac{2}{3}\right)^{n} \left( \prod_{i=1}^{n} X_{i}\right) \displaystyle e^{\sum_{i=1}^{n} X_{i}}(32​)n(∏i=1n​Xi​)e∑i=1n​Xi​
  • \left( \frac{2}{3}\right)^{2n} \displaystyle e^{\sum_{i=1}^{n} X_{i}}(32​)2ne∑i=1n​Xi​
  • \left( \frac{2}{3}\right)^{2n} \displaystyle e^{-\sum_{i=1}^{n} X_{i}}(32​)2ne−∑i=1n​Xi​

Q4. Suppose that X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ is a random sample from the continuous uniform distribution over the interval from 00 to 11, which has pdf

  • ​f(x) =
  • ⎧⎩⎨1θ0,,0<x<θotherwise
  • ​f(x)={θ1​0​,,​0<x<θotherwise​
  • Find the pdf for the maximum value in the sample.
  • f(x) =
  • ⎧⎩⎨1θn0,,0<x<θotherwise
  • f(x)={θn1​0​,,​0<x<θotherwise​
  • f(x) =
  • ⎧⎩⎨1−(1−xθ)n0,,0<x<θotherwise
  • f(x)={1−(1−θx​)n0​,,​0<x<θotherwise​
  • f(x) =
  • ⎧⎩⎨(xθ)n0,,0<x<θotherwise
  • f(x)={(θx​)n0​,,​0<x<θotherwise​
  • f(x) =
  • {nθnxn−10,,0<x<θotherwise
  • f(x)={θnn​xn−10​,,​0<x<θotherwise​

Q5. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample from the N(\mu,\sigma^{2})N(μ,σ2) where \muμ is known.

On which statistic is the best test of

H_{0}: \sigma^{2} = \sigma_{0}^{2} \,\,\, \text{versus} \,\,\, H_{1}: \sigma^{2}>\sigma_{0}^{2}H0​:σ2=σ02​versusH1​:σ2>σ02​

based?

  • \overline{X}X
  • \sum_{i=1}^{n} (X_{i}-\mu)^{2}∑i=1n​(Xi​−μ)2
  • \sum_{i=1}^{n} X_{i}∑i=1n​Xi​
  • \sum_{i=1}^{n} X_{i}^{2}∑i=1n​Xi2​

Quiz 2: Uniformly Most Powerful Tests and F-Tests

Q1. A teacher believes that the standard deviation of scores for a particular Midterm that he gives every semester is 4 points. His current students claim that the standard deviation is more than 4 points.

L​et \sigma^{2}σ2 be the true variance for the Midterm scores.

A​ random sample of 10 Midterm scores from the current semester has an observed standard deviation of s=5.2s=5.2 points.

Assuming that test scores are normally distributed, c​onsider testing the hypotheses

H_{0}: \sigma^{2}=16 \,\,\, \text{versus} \,\,\, H_{1}: \sigma^{2}>16H0​:σ2=16versusH1​:σ2>16.

G​ive the P-value for the appropriate test, rounded to 3 decimal places.

Q3. C​onsider a random sample X_{1}X1​, of size 1, from the continuous distribution with pdf

f(x;\theta)=

⎧⎩⎨1−θ2(x−12),0,0<x<1otherwise

f(x;θ)={1−θ2(x−21​),0,​0<x<1otherwise​

f​or some parameter -1<\theta< 1−1<θ<1.

D​oes a uniformly most powerful (UMP) test for

H_{0}: \theta = 0 \,\,\, \text{versus} \,\,\, H_{1}: \theta \ne 0H0​:θ=0versusH1​:θ​=0

e​xist?

  • Y​es, and the form of the test will be to reject H_{0}H0​, in favor of H_{1}H1​, if X_{1} \geq cX1​≥c OR X_{1} \leq -cX1​≤−c for some constant c>0c>0 to be determined.
  • Y​es, and the form of the test will be to reject H_{0}H0​, in favor of H_{1}H1​, if X_{1} \geq cX1​≥c for some constant cc to be determined.
  • Y​es, and the form of the test will be to reject H_{0}H0​, in favor of H_{1}H1​, if X_{1} \leq cX1​≤c for some constant cc to be determined.
  • N​o. UMP tests never exist for two-sided alternative hypotheses.

Q3. A​ random sample of size 88 from the N(\mu_{1},\sigma_{1}^{2})N(μ1​,σ12​) distribution has a sample variance of s_{1}^{2}=13.2s12​=13.2.

  • An independent​ random sample of size 66 from a N(\mu_{1},\sigma_{2}^{2})N(μ1​,σ22​) has a sample variance of s_{2}^{2}=15.1s22​=15.1.
  • I​s there evidence to suggest that \sigma_{1}^{2} \ne \sigma_{2}^{2}σ12​​=σ22​?
  • U​se \alpha=0.03α=0.03 level of significance.
  • N​o. The test statistic is approximately 1.14391.1439 and we would reject the null hypothesis that the variances are equal, in favor of the stated alternative, if it is greater than 6.28006.2800.
  • N​o. The test statistic is approximately 1.14391.1439 and we would reject the null hypothesis that the variances are equal, in favor of the stated alternative, if it is greater than 6.42876.4287.
  • N​o. The test statistic is approximately 1.14391.1439 and we would reject the null hypothesis that the variances are equal, in favor of the stated alternative, if it is greater than 4.91504.9150.
  • N​o. The test statistic is approximately 1.14391.1439 and we would reject the null hypothesis that the variances are equal, in favor of the stated alternative, if it is greater than 4.35304.3530.

Q4. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample from the N(\mu,1)N(μ,1).

Consider deriving the uniformly most powerful (UMP) test for

H_{0}: \mu=\mu_{0} \,\,\, \text{versus} \,\,\, H_{1}: \mu>\mu_{0}.H0​:μ=μ0​versusH1​:μ>μ0​.

W​e begin by finding the best test for

H_{0}: \mu=\mu_{0} \,\,\, \text{versus} \,\,\, H_{1}: \mu = \mu_{1},H0​:μ=μ0​versusH1​:μ=μ1​,

w​here \mu_{1}μ1​ is some fixed value that is greater than \mu_{0}μ0​.

W​hich rejection rules will give you the UMP test?

  • Test A: R​eject H_{0}H0​, n favor of H_{1}H1​ if
  • \exp \left[ -\frac{1}{2} \left( \sum_{i=1}^{n} (X_{i}-\mu_{0})^{2} – \sum_{i=1}^{n} (X_{i}-\mu_{1})^{2} \right) \right] \leq kexp[−21​(∑i=1n​(Xi​−μ0​)2−∑i=1n​(Xi​−μ1​)2)]≤k
  • f​or some kk to be determined.
  • Test B: R​eject H_{0}H0​, n favor of H_{1}H1​ if
  • \sum_{i=1}^{n} (X_{i}-\mu_{0})^{2} – \sum_{i=1}^{n} (X_{i}-\mu_{1})^{2} \geq k∑i=1n​(Xi​−μ0​)2−∑i=1n​(Xi​−μ1​)2≥k
  • f​or some kk to be determined.
  • B​oth tests A and B above.
  • N​one of the above.

Q5. True or False. F​or a simple H_{0}H0​ and simple H_{1}H1​, the uniformly most powerful test of size \alphaα is the same as the best test.

  • T​rue
  • F​alse

Week 5 Quiz Answers

Quiz 1: Adventures in GLRTs

Q1. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample of size nn from the continous distribution with pdf

f(x;\theta) =

{θ(1−x)θ−1,0,0<x<1otherwise

f(x;θ)={θ(1−x)θ−1,0,​0<x<1otherwise​

f​or some parameter \theta>0θ>0.

C​onsider testing

H_{0}: \theta=1 \,\,\, \text{versus} \,\,\, H_{1}: \theta \ne 1.H0​:θ=1versusH1​:θ​=1.

W​hich of the following statements are true? (Check all that apply.)

  • T​he uniformly most powerful (UMP) test is the same as the generalized likelihood ratio test (GLRT).
  • A​ uniformly most powerful (UMP) test does not exist.
  • U​nder H_{0}H0​, X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ are iid from the uniform distribution over the interval (0,1)(0,1).
  • F​or a large sample size, the generalized likelihood ratio \lambda(\vec{X})λ(X) has an approximate \chi^{2}(1)χ2(1) distribution.

Q2. L​et X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ be a random sample of size nn from the continous distribution with pdf

f(x;\theta) =

{θ(1−x)θ−1,0,0<x<1otherwise

f(x;θ)={θ(1−x)θ−1,0,​0<x<1otherwise​

f​or some parameter \theta>0θ>0.

C​onsider testing

H_{0}: \theta=1 \,\,\, \text{versus} \,\,\, H_{1}: \theta \ne 1H0​:θ=1versusH1​:θ​=1.

W​hat is the maximum likelihood estimator needed for computing the denominator of the generalized likelihood ratio (GLR)?

  • \widehat{\theta} = \frac{n}{\sum_{i=1}^{n} (1-X_{i})}θ=∑i=1n​(1−Xi​)n​
  • \widehat{\theta} = \frac{\prod_{i=1}^{n} (1-X_{i})}{n}θ=n∏i=1n​(1−Xi​)​
  • \widehat{\theta} = \frac{-n}{\sum_{i=1}^{n} \ln(1-X_{i})}θ=∑i=1n​ln(1−Xi​)−n​
  • \widehat{\theta} = \overline{X}θ=X

Q3. True or False.

A​ generalized likelihood ratio test will always be less powerful than a UMP test.

  • T​rue
  • F​alse

Q4. T​rue or False.

W​hen computing a generalized likelihood ratio (GLR) for some H_{0}H0​ versus some H_{1}H1​, the restricted MLE is always less than or equal to the unrestricted MLE.

  • T​rue
  • F​alse

Q5. S​uppose that X_{1}, X_{2}, \ldots, X_{n}X1​,X2​,…,Xn​ is a random sample from the \Gamma(2,\beta)Γ(2,β) distribution . Suppose that nn is “large”.

C​onsider testing

H_{0}: \beta = 1 \,\,\, \text{versus} \,\,\, H_{1}: \beta \ne 1H0​:β=1versusH1​:β​=1

using an approximate large sample generalized likelihood ratio test.

S​uppose that the generalized likelihood ratio \lambda(\vec{X})λ(X) is observed to be \lambda(\vec{x})=0.67λ(x)=0.67.

W​hich of the following is an approximate large sample test using \alpha=0.05α=0.05?

  • I​ can not answer this question without knowing the value of nn.
  • The rule is to r​eject H_{0}H0​, in favor of H_{1}H1​ if -2 \ln \lambda(\vec{X})>0.0039−2lnλ(X)>0.0039. For the given \lambda(\vec{x})λ(x), we reject H_{0}H0​.
  • The rule is to reject H_{0}H0​, in favor of H_{1}H1​ if -2 \ln \lambda(\vec{X})>5.0239−2lnλ(X)>5.0239. For the given \lambda(\vec{x})λ(x), we fail to reject H_{0}H0​.
  • The rule is to reject H_{0}H0​, in favor of H_{1}H1​ if -2 \ln \lambda(\vec{X})>3.841−2lnλ(X)>3.841. For the given \lambda(\vec{x})λ(x), we fail to reject H_{0}H0​.

Conclusion

Hopefully, this article will be useful for you to find all the Week, final assessment, and Peer Graded Assessment Answers of Statistical Inference and Hypothesis Testing in Data Science Applications Quiz of Coursera and grab some premium knowledge with less effort. If this article really helped you in any way then make sure to share it with your friends on social media and let them also know about this amazing training. You can also check out our other course Answers. So, be with us guys we will share a lot more free courses and their exam/quiz solutions also, and follow our Techno-RJ Blog for more updates.

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  1502. El Balanceo de Componentes: Elemento Clave para un Desempeño Óptimo

    ¿ En algún momento te has dado cuenta de movimientos irregulares en una máquina? ¿O tal vez escuchaste ruidos anómalos? Muchas veces, el problema está en algo tan básico como una irregularidad en un componente giratorio . Y créeme, ignorarlo puede costarte bastante dinero .

    El equilibrado de piezas es un paso esencial en la construcción y conservación de maquinaria agrícola, ejes, volantes y elementos de motores eléctricos. Su objetivo es claro: evitar vibraciones innecesarias que pueden causar daños serios a largo plazo .

    ¿Por qué es tan importante equilibrar las piezas?
    Imagina que tu coche tiene un neumático con peso desigual. Al acelerar, empiezan las sacudidas, el timón vibra y resulta incómodo circular así. En maquinaria industrial ocurre algo similar, pero con consecuencias mucho más graves :

    Aumento del desgaste en bearings y ejes giratorios
    Sobrecalentamiento de partes críticas
    Riesgo de colapsos inesperados
    Paradas imprevistas que exigen arreglos costosos
    En resumen: si no se corrige a tiempo, una leve irregularidad puede transformarse en un problema grave .

    Métodos de equilibrado: cuál elegir
    No todos los casos son iguales. Dependiendo del tipo de pieza y su uso, se aplican distintas técnicas:

    Equilibrado dinámico
    Recomendado para componentes que rotan rápidamente, por ejemplo rotores o ejes. Se realiza en máquinas especializadas que detectan el desequilibrio en múltiples superficies . Es el método más exacto para asegurar un movimiento uniforme .
    Equilibrado estático
    Se usa principalmente en piezas como ruedas, discos o volantes . Aquí solo se corrige el peso excesivo en una sola superficie . Es ágil, práctico y efectivo para determinados sistemas.
    Corrección del desequilibrio: cómo se hace
    Taladrado selectivo: se perfora la región con exceso de masa
    Colocación de contrapesos: como en ruedas o anillos de volantes
    Ajuste de masas: típico en bielas y elementos estratégicos
    Equipos profesionales para detectar y corregir vibraciones
    Para hacer un diagnóstico certero, necesitas herramientas precisas. Hoy en día hay opciones económicas pero potentes, tales como:

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  1509. Equilibrado de piezas
    El Equilibrado de Piezas: Clave para un Funcionamiento Eficiente

    ¿ Has percibido alguna vez temblores inusuales en un equipo industrial? ¿O sonidos fuera de lo común? Muchas veces, el problema está en algo tan básico como una irregularidad en un componente giratorio . Y créeme, ignorarlo puede costarte más de lo que imaginas.

    El equilibrado de piezas es una tarea fundamental tanto en la fabricación como en el mantenimiento de maquinaria agrícola, ejes, volantes, rotores y componentes de motores eléctricos . Su objetivo es claro: prevenir movimientos indeseados capaces de generar averías importantes con el tiempo .

    ¿Por qué es tan importante equilibrar las piezas?
    Imagina que tu coche tiene una llanta mal nivelada . Al acelerar, empiezan los temblores, el manubrio se mueve y hasta puede aparecer cierta molestia al manejar . En maquinaria industrial ocurre algo similar, pero con consecuencias mucho más graves :

    Aumento del desgaste en bearings y ejes giratorios
    Sobrecalentamiento de partes críticas
    Riesgo de colapsos inesperados
    Paradas imprevistas que exigen arreglos costosos
    En resumen: si no se corrige a tiempo, un pequeño desequilibrio puede convertirse en un gran dolor de cabeza .

    Métodos de equilibrado: cuál elegir
    No todos los casos son iguales. Dependiendo del tipo de pieza y su uso, se aplican distintas técnicas:

    Equilibrado dinámico
    Recomendado para componentes que rotan rápidamente, por ejemplo rotores o ejes. Se realiza en máquinas especializadas que detectan el desequilibrio en múltiples superficies . Es el método más exacto para asegurar un movimiento uniforme .
    Equilibrado estático
    Se usa principalmente en piezas como ruedas, discos o volantes . Aquí solo se corrige el peso excesivo en una única dirección. Es ágil, práctico y efectivo para determinados sistemas.
    Corrección del desequilibrio: cómo se hace
    Taladrado selectivo: se perfora la región con exceso de masa
    Colocación de contrapesos: tal como en neumáticos o perfiles de poleas
    Ajuste de masas: típico en bielas y elementos estratégicos
    Equipos profesionales para detectar y corregir vibraciones
    Para hacer un diagnóstico certero, necesitas herramientas precisas. Hoy en día hay opciones disponibles y altamente productivas, por ejemplo :

    ✅ Balanset-1A — Tu asistente móvil para analizar y corregir oscilaciones

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  1511. Balanceo móvil en campo:
    Respuesta inmediata sin mover equipos

    Imagina esto: tu rotor empieza a temblar, y cada minuto de inactividad genera pérdidas. ¿Desmontar la máquina y esperar días por un taller? Descartado. Con un equipo de equilibrado portátil, resuelves sobre el terreno en horas, sin mover la maquinaria.

    ¿Por qué un equilibrador móvil es como un “paquete esencial” para máquinas rotativas?
    Fácil de transportar y altamente funcional, este dispositivo es el recurso básico en cualquier intervención. Con un poco de práctica, puedes:
    ✅ Corregir vibraciones antes de que dañen otros componentes.
    ✅ Evitar paradas prolongadas, manteniendo la producción activa.
    ✅ Operar en zonas alejadas, ya sea en instalaciones marítimas o centrales solares.

    ¿Cuándo es ideal el equilibrado rápido?
    Siempre que puedas:
    – Acceder al rotor (eje, ventilador, turbina, etc.).
    – Colocar sensores sin interferencias.
    – Modificar la distribución de masa (agregar o quitar contrapesos).

    Casos típicos donde conviene usarlo:
    La máquina muestra movimientos irregulares o ruidos atípicos.
    No hay tiempo para desmontajes (producción crítica).
    El equipo es de alto valor o esencial en la línea de producción.
    Trabajas en áreas donde no hay asistencia mecánica disponible.

    Ventajas clave vs. llamar a un técnico
    | Equipo portátil | Servicio externo |
    |—————-|——————|
    | ✔ Sin esperas (acción inmediata) | ❌ Retrasos por programación y transporte |
    | ✔ Monitoreo preventivo (evitas fallas mayores) | ❌ Suele usarse solo cuando hay emergencias |
    | ✔ Reducción de costos operativos con uso continuo | ❌ Costos recurrentes por servicios |

    ¿Qué máquinas se pueden equilibrar?
    Cualquier sistema rotativo, como:
    – Turbinas de vapor/gas
    – Motores industriales
    – Ventiladores de alta potencia
    – Molinos y trituradoras
    – Hélices navales
    – Bombas centrífugas

    Requisito clave: acceso suficiente para medir y corregir el balance.

    Tecnología que simplifica el proceso
    Los equipos modernos incluyen:
    Aplicaciones didácticas (para usuarios nuevos o técnicos en formación).
    Evaluación continua (informes gráficos comprensibles).
    Batería de larga duración (perfecto para zonas remotas).

    Ejemplo práctico:
    Un molino en una mina mostró movimientos inusuales. Con un equipo portátil, el técnico localizó el error rápidamente. Lo corrigió añadiendo contrapesos y evitó una parada de 3 días.

    ¿Por qué esta versión es más efectiva?
    – Estructura más dinámica: Formato claro ayuda a captar ideas clave.
    – Enfoque práctico: Incluye casos ilustrativos y contrastes útiles.
    – Lenguaje persuasivo: Frases como “kit de supervivencia” o “minimizas riesgos importantes” refuerzan el valor del servicio.
    – Detalles técnicos útiles: Se especifican requisitos y tecnologías modernas.

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  1514. Servicio de Equilibrado
    ¿Oscilaciones inusuales en tu máquina? Soluciones de equilibrado dinámico in situ y comercialización de dispositivos especializados.

    ¿Has notado movimientos extraños, zumbidos inesperados o degradación rápida en tus equipos? Esto indica claramente de que tu equipo industrial necesita un balanceo dinámico experto.

    En lugar de desmontar y enviar tus máquinas a un taller, nosotros vamos hasta tu planta industrial con equipos de última generación para solucionar la falla sin interrumpir tu producción.

    Beneficios de nuestro ajuste en planta
    ✔ Evitamos desarmados y transportes — Trabajamos directamente en tus instalaciones.
    ✔ Diagnóstico preciso — Empleamos dispositivos de alta precisión para identificar el problema.
    ✔ Resultados inmediatos — Soluciones rápidas en cuestión de horas.
    ✔ Reporte completo — Certificamos el proceso con datos comparativos.
    ✔ Conocimiento en diversos sectores — Atendemos desde grandes turbinas hasta motores compactos.

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  1515. Equilibrio in situ
    El Equilibrado de Piezas: Clave para un Funcionamiento Eficiente

    ¿Alguna vez has notado vibraciones extrañas en una máquina? ¿O tal vez ruidos que no deberían estar ahí? Muchas veces, el problema está en algo tan básico como un desequilibrio en alguna pieza rotativa . Y créeme, ignorarlo puede costarte caro .

    El equilibrado de piezas es un procedimiento clave en la producción y cuidado de equipos industriales como ejes, volantes, rotores y partes de motores eléctricos . Su objetivo es claro: impedir oscilaciones que, a la larga, puedan provocar desperfectos graves.

    ¿Por qué es tan importante equilibrar las piezas?
    Imagina que tu coche tiene una llanta mal nivelada . Al acelerar, empiezan los temblores, el manubrio se mueve y hasta puede aparecer cierta molestia al manejar . En maquinaria industrial ocurre algo similar, pero con consecuencias considerablemente más serias:

    Aumento del desgaste en bearings y ejes giratorios
    Sobrecalentamiento de partes críticas
    Riesgo de fallos mecánicos repentinos
    Paradas imprevistas que exigen arreglos costosos
    En resumen: si no se corrige a tiempo, una mínima falla podría derivar en una situación compleja.

    Métodos de equilibrado: cuál elegir
    No todos los casos son iguales. Dependiendo del tipo de pieza y su uso, se aplican distintas técnicas:

    Equilibrado dinámico
    Ideal para piezas que giran a alta velocidad, como rotores o ejes . Se realiza en máquinas especializadas que detectan el desequilibrio en múltiples superficies . Es el método más exacto para asegurar un movimiento uniforme .
    Equilibrado estático
    Se usa principalmente en piezas como neumáticos, discos o volantes de inercia. Aquí solo se corrige el peso excesivo en una única dirección. Es rápido, fácil y funcional para algunos equipos .
    Corrección del desequilibrio: cómo se hace
    Taladrado selectivo: se elimina material en la zona más pesada
    Colocación de contrapesos: por ejemplo, en llantas o aros de volantes
    Ajuste de masas: común en cigüeñales y otros componentes críticos
    Equipos profesionales para detectar y corregir vibraciones
    Para hacer un diagnóstico certero, necesitas herramientas precisas. Hoy en día hay opciones económicas pero potentes, tales como:

    ✅ Balanset-1A — Tu asistente móvil para analizar y corregir oscilaciones

    Reply

  1516. analizador de vibrasiones
    Balanceo móvil en campo:
    Respuesta inmediata sin mover equipos

    Imagina esto: tu rotor empieza a temblar, y cada minuto de inactividad genera pérdidas. ¿Desmontar la máquina y esperar días por un taller? Olvídalo. Con un equipo de equilibrado portátil, resuelves sobre el terreno en horas, sin mover la maquinaria.

    ¿Por qué un equilibrador móvil es como un “herramienta crítica” para máquinas rotativas?
    Compacto, adaptable y potente, este dispositivo es una pieza clave en el arsenal del ingeniero. Con un poco de práctica, puedes:
    ✅ Prevenir averías mayores al detectar desbalances.
    ✅ Evitar paradas prolongadas, manteniendo la producción activa.
    ✅ Operar en zonas alejadas, ya sea en instalaciones marítimas o centrales solares.

    ¿Cuándo es ideal el equilibrado rápido?
    Siempre que puedas:
    – Contar con visibilidad al sistema giratorio.
    – Colocar sensores sin interferencias.
    – Realizar ajustes de balance mediante cambios de carga.

    Casos típicos donde conviene usarlo:
    La máquina presenta anomalías auditivas o cinéticas.
    No hay tiempo para desmontajes (proceso vital).
    El equipo es costoso o difícil de detener.
    Trabajas en zonas remotas sin infraestructura técnica.

    Ventajas clave vs. llamar a un técnico
    | Equipo portátil | Servicio externo |
    |—————-|——————|
    | ✔ Rápida intervención (sin demoras) | ❌ Demoras por agenda y logística |
    | ✔ Mantenimiento proactivo (previenes daños serios) | ❌ Suele usarse solo cuando hay emergencias |
    | ✔ Ahorro a largo plazo (menos desgaste y reparaciones) | ❌ Costos recurrentes por servicios |

    ¿Qué máquinas se pueden equilibrar?
    Cualquier sistema rotativo, como:
    – Turbinas de vapor/gas
    – Motores industriales
    – Ventiladores de alta potencia
    – Molinos y trituradoras
    – Hélices navales
    – Bombas centrífugas

    Requisito clave: espacio para instalar sensores y realizar ajustes.

    Tecnología que simplifica el proceso
    Los equipos modernos incluyen:
    Software fácil de usar (con instrucciones visuales y automatizadas).
    Análisis en tiempo real (gráficos claros de vibraciones).
    Batería de larga duración (perfecto para zonas remotas).

    Ejemplo práctico:
    Un molino en una mina empezó a generar riesgos estructurales. Con un equipo portátil, el técnico detectó un desbalance en 20 minutos. Lo corrigió añadiendo contrapesos y ahorró jornadas de inactividad.

    ¿Por qué esta versión es más efectiva?
    – Estructura más dinámica: Listas, tablas y negritas mejoran la legibilidad.
    – Enfoque práctico: Ofrece aplicaciones tangibles del método.
    – Lenguaje persuasivo: Frases como “kit de supervivencia” o “previenes consecuencias críticas” refuerzan el valor del servicio.
    – Detalles técnicos útiles: Se especifican requisitos y tecnologías modernas.

    ¿Necesitas ajustar el tono (más comercial) o añadir keywords específicas? ¡Aquí estoy para ayudarte! ️

    Reply

  1517. ¡Vendemos máquinas para balanceo!
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