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About Statistical Inference Course
In statistical inference, one uses data to infer information about a population or a scientific fact. Statistical modeling, data-oriented techniques, and the explicit use of designs and randomization in studies are just a few examples of the many ways in which inference can be carried out.
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Statistical Inference Quiz Answers
Statistical Inference Week 1 Quiz Answers
Question 1)
Consider influenza epidemics for two-parent heterosexual families.
Suppose that the probability is 17% that at least one of the parents has contracted the disease.
The probability that the father has contracted influenza is 12% while the probability that both the mother and father have contracted the disease is 6%.
What is the probability that the mother has contracted influenza?
(Hints look at lecture 2 around 5:30 and homework question on page 3/10).
- 6%
- 17%
- 11%
- 5%
Question 2)
A random variable, X is uniform, a box from 0 to 1 of height 1.
(So that its density is f(x)=1 for 0≤x≤1.) What is its 75th percentile?
(Hints, look at lecture 2 around 21:30 and homework 1 page 4/10. Also, look up the help function for the qunif command in R.)
- 0.10
- 0.25
- 0.75
- 0.50
Question 3)
You are playing a game with a friend where you flip a coin and if it comes up heads you give her X dollars and if it comes up tails she gives you Y dollars. The probability that the coin is heads is p (some number between 0 and 1.) What has to be true about X and Y to make so that both of your expected total earnings is 0. The game would then be called “fair”.
(Hints, look at Lecture 4 from 0 to 6:50 and Homework 1 page 5/10. Also, for further reading on fair games and gambling, start with the Dutch Book problem ).
- X=Y
- p=X/Y
- p/1−p=Y/X
- p/1−p=X/Y
Question 4)
A density that looks like a normal density (but may or may not be exactly normal) is exactly symmetric about zero. (Symmetric means if you flip it around zero it looks the same.) What is its median?
(Hints, look at quantiles from Lecture 2 around 21:30 and the problem on page 9/10 from Homework 1.)
- The median must be 0.
- The median must be different from the mean.
- The median must be 1.
- We can’t conclude anything about the median.
Question 5)
Consider the following PMF shown below in R.
x <- 1:4
p <- x/sum(x)
temp <- rbind(x, p)
rownames(temp) <- c(“X”, “Prob”)
temp
## [,1] [,2] [,3] [,4]
## X 1.0 2.0 3.0 4.0
## Prob 0.1 0.2 0.3 0.4
What is the mean?
(Hint, watch Lecture 4 on expectations of PMFs and look at the Homework 1 problem on page 10/10 for a similar problem calculating the variance.)
- 4
- 2
- 1
- 3
Question 6)
A web site (www.medicine.ox.ac.uk/bandolier/band64/b64-7.html) for home pregnancy tests cites the following:
“When the subjects using the test were women who collected and tested their own samples, the overall sensitivity was 75%. Specificity was also low, in the range 52% to 75%.” Assume the lower value for the specificity.
Suppose a subject has a positive test and that 30% of women taking pregnancy tests are actually pregnant.
What number is closest to the probability of pregnancy given the positive test?
(Hints, watch Lecture 3 at around 7 minutes for a similar example. Also, there’s a lot of Bayes’ rule problems and descriptions out there, for example here’s one for HIV testing. Note, discussions of Bayes’ rule can get pretty heady. So if it’s new to you, stick to basic treatments of the problem. Also see Homework 2 question on page 5/12.)
- 20%
- 30%
- 40%
- 10%
Statistical Inference Week 2 Quiz Answers
Question 1)
What is the variance of the distribution of the average an IID draw of n observations from a population with mean μ and variance σ2.
Your Answer Score Explanation
- σ2
- σ/n
- 2σ/n‾‾√
- σ2/n
Question 2)
Suppose that diastolic blood pressures (DBPs) for men aged 35-44 are normally distributed with a mean of 80 (mm Hg) and a standard deviation of 10. About what is the probability that a random 35-44 year old has a DBP less than 70?
- 16%
- 8%
- 22%
- 32%
Question 3)
Brain volume for adult women is normally distributed with a mean of about 1,100 cc for women with a standard deviation of 75 cc. What brain volume represents the 95th percentile?
- approximately 1175
- approximately 977
- approximately 1223
- approximately 1247
Question 4)
Refer to the previous question. Brain volume for adult women is about 1,100 cc for women with a standard deviation of 75 cc. Consider the sample mean of 100 random adult women from this population. What is the 95th percentile of the distribution of that sample mean?
- approximately 1112 cc
- approximately 1115 cc
- approximately 1088 cc
- approximately 1110 cc
Question 5)
#You flip a fair coin 5 times, about what’s the probability of getting 4 or 5 heads?
- 19%
- 3%
- 6%
- 12%
Question 6)
The respiratory disturbance index (RDI), a measure of sleep disturbance, for a specific population has a mean of 15 (sleep events per hour) and a standard deviation of 10. They are not normally distributed. Give your best estimate of the probability that a sample mean RDI of 100 people is between 14 and 16 events per hour?
- 68%
- 95%
- 34%
- 47.5%
Question 7)
Consider a standard uniform density. The mean for this density is .5 and the variance is 1 / 12. You sample 1,000 observations from this distribution and take the sample mean, what value would you expect it to be near?
- 0.75
- 0.10
- 0.25
- 0.5
Question 8)
#The number of people showing up at a bus stop is assumed to be Poisson with a mean of 5 people per hour. You watch the bus stop for 3 hours. About what’s the probability of viewing 10 or fewer people?
- 0.03
- 0.08
- 0.12
- 0.06
Statistical Inference Week 3 Quiz Answers
Question 1)
In a population of interest, a sample of 9 men yielded a sample average brain volume of 1,100cc and a standard deviation of 30cc. What is a 95% Students T confidence interval for the mean brain volume in this new population?
- [1092, 1108]
- [1080, 1120]
- [1031, 1169]
- [1077, 1123]
Question 2)
A diet pill is given to 9 subjects over six weeks. The average difference in weight (follow up – baseline) is -2 pounds.
What would the standard deviation of the difference in weight have to be for the upper endpoint of the 95% T confidence interval to touch 0?
- 2.60
- 1.50
- 2.10
- 0.30
Question 3
In an effort to improve running performance, 5 runners were either given a protein supplement or placebo.
Then, after a suitable washout period, they were given the opposite treatment.
Their mile times were recorded under both the treatment and placebo, yielding 10 measurements with 2 per subject.
The researchers intend to use a T test and interval to investigate the treatment.
Should they use a paired or independent group T test and interval?
- #It’s necessary to use both
- #A paired interval
- #You could use either
- #Independent groups, since all subjects were seen under both systems
Question 4)
In a study of emergency room waiting times, investigators consider a new and the standard triage systems.
To test the systems, administrators selected 20 nights and randomly assigned the new triage system to be used on 10 nights and the standard system on the remaining 10 nights. They calculated the nightly median waiting time (MWT) to see a physician.
The average MWT for the new system was 3 hours with a variance of 0.60 while the average MWT for the old system was 5 hours with a variance of 0.68. Consider the 95% confidence interval estimate for the differences of the mean MWT associated with the new system.
Assume a constant variance. What is the interval? Subtract in this order (New System – Old System).
- [1.29, 2.70]
- [1.25, 2.75]
- [-2,70, -1.29]
- [-2.75, -1.25]
Question 5)
Suppose that you create a 95% T confidence interval. You then create a 90% interval using the same data. What can be said about the 90% interval with respect to the 95% interval?
- It is impossible to tell.
- The interval will be narrower.
- The interval will be wider
- The interval will be the same width, but shifted.
Question 6)
To further test the hospital triage system, administrators selected 200 nights and randomly assigned a new triage system to be used on 100 nights and a standard system on the remaining 100 nights. They calculated the nightly median waiting time (MWT) to see a physician. The average MWT for the new system was 4 hours with a standard deviation of 0.5 hours while the average MWT for the old syste was 6 hours with a standard deviation of 2 hours.
Consider the hypothesis of a decrease in the mean MWT associated with the new treatment.
What does the 95% independent group confidence interval with unequal variances suggest vis a vis this hypothesis?
#(Because there’s so many observations per group, just use the Z quantile instead of the T.)
- When subtracting (old – new) the interval is entirely above zero. The new system appears to be effective.
- When subtracting (old – new) the interval contains 0. The new system appears to be effective.
- When subtracting (old – new) the interval is entirely above zero. The new system does not appear to be effective.
- When subtracting (old – new) the interval contains 0. There is not evidence suggesting that the new system is effective.
Question 7)
Suppose that 18 obese subjects were randomized, 9 each, to a new diet pill and a placebo.
Subjects’ body mass indices (BMIs) were measured at a baseline and again after having received the treatment or placebo for four weeks.
The average difference from follow-up to the baseline (followup – baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the placebo group. The corresponding standard deviations of the differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for the placebo group. Does the change in BMI over the four week period appear to differ between the treated and placebo groups?
Assuming normality of the underlying data and a common population variance, calculate the relevant *90%* t confidence interval.
Subtract in the order of (Treated – Placebo) with the smaller (more negative) number first.
- [-5.364, -2.636]
- [-5.531, -2.469]
- [2.469, 5.531]
- [2.636, 5.364]
Statistical Inference Week 1 Quiz Answers
Question 1)
A pharmaceutical company is interested in testing a potential blood pressure lowering medication.
Their first examination considers only subjects that received the medication at baseline then two weeks later.
The data are as follows (SBP in mmHg)
Subject Baseline Week 2
1 140 132
2 138 135
3 150 151
4 148 146
5 135 130
Consider testing the hypothesis that there was a mean reduction in blood pressure? Give the P-value for the associated two sided T test.
(Hint, consider that the observations are paired.)
- 0.05
- 0.10
- 0.087
- 0.043
Question 2)
A sample of 9 men yielded a sample average brain volume of 1,100cc and a standard deviation of 30cc.
What is the complete set of values of μ0 that a test of H0:μ=μ0 would fail to reject the null hypothesis in a two sided 5% Students t-test?
- 1031 to 1169
- 1077 to 1123
- 1081 to 1119
- 1080 to 1120
Question 3)
Researchers conducted a blind taste test of Coke versus Pepsi.
Each of four people was asked which of two blinded drinks given in random order that they preferred.
The data was such that 3 of the 4 people chose Coke.
Assuming that this sample is representative, report a P-value for a test of the hypothesis that Coke is preferred to Pepsi using a one sided exact test.
- 0.62
- 0.005
- 0.10
- 0.31
Question 4)
Infection rates at a hospital above 1 infection per 100 person days at risk are believed to be too high and are used as a benchmark.
A hospital that had previously been above the benchmark recently had 10 infections over the last 1,787 person days at risk.
About what is the one sided P-value for the relevant test of whether the hospital is *below* the standard?
- 0.11
- 0.03
- 0.52
- 0.22
Question 5)
Suppose that 18 obese subjects were randomized, 9 each, to a new diet pill and a placebo.
Subjects’ body mass indices (BMIs) were measured at a baseline and again after having received the treatment or placebo for four weeks.
The average difference from follow-up to the baseline (followup – baseline) was −3 kg/m2 for the treated group and 1 kg/m2 for the placebo group. The corresponding standard deviations of the differences was 1.5 kg/m2 for the treatment group and 1.8 kg/m2 for the placebo group. Does the change in BMI appear to differ between the treated and placebo groups?
Assuming normality of the underlying data and a common population variance, give a pvalue for a two sided t test.
- Less than 0.01
- Less than 0.10 but larger than 0.05
- Larger than 0.10
- Less than 0.05, but larger than 0.01
Question 6)
Brain volumes for 9 men yielded a 90% confidence interval of 1,077 cc to 1,123 cc.
Would you reject in a two sided 5% hypothesis test of H0:μ=1,078?
- #No you wouldn’t reject.
- #Yes you would reject.
- #It’s impossible to tell.
- #Where does Brian come up with these questions?
Question 7)
Researchers would like to conduct a study of 100 healthy adults to detect a four year mean brain volume loss of .01 mm3.
Assume that the standard deviation of four year volume loss in this population is .04 mm3.
About what would be the power of the study for a 5% one sided test versus a null hypothesis of no volume loss?
- 0.60
- 0.80
- 0.70
- 0.50
Question 8)
Researchers would like to conduct a study of n healthy adults to detect a four year mean brain volume loss of .01 mm3.
Assume that the standard deviation of four year volume loss in this population is .04 mm3.
About what would be the value of n needded for 90% power of type one error rate of 5% one sided test versus a null hypothesis of no volume loss?
- 180
- 160
- 120
- 140
Question 9)
As you increase the type one error rate, α, what happens to power?
- #It’s impossible to tell given the information in the problem.
- #You will get smaller power.
- #No, for real, where does Brian come up with these problems?
- #You will get larger power.
More About This Course
In statistical inference, one uses data to infer information about a population or a scientific fact. Statistical modeling, data-oriented techniques, and the explicit use of designs and randomization in studies are just a few examples of the many ways in which inference can be carried out.
Furthermore, there are several difficulties (missing data, observed and unobserved confounding, biases) and various theories (frequentists, Bayesian, likelihood, design based, etc.) for carrying out inference. It’s not uncommon for a practitioner to feel completely lost in the tangled web of available methods, ideas, and subtleties.
Using a hands-on, action-oriented methodology, this course introduces the principles of inference.
By the end of the course, students will be able to apply the broad principles of statistical inference to their own data analysis decisions.
Several different courses make up this one.
You can use this class toward several different Specializations or Professional Certificates. Any of the following credentials can be augmented by successful completion of this course:
- Concentration in Data Science: Statistics and Machine Learning
- Expertise in Data Science
WHAT YOU WILL BE TAKING AWAY
- Learn how to extrapolate from sample data to generalizable generalisations about populations or scientific truths.
- Specify the range of possible values, their boundaries, and the confidence intervals you’re comfortable with
- Make use of p-values, CIs, and permutation tests
- Take calculated risks with your data analysis
SKILLS YOU WILL GAIN
- Statistics
- Statistical Inference
- Statistical Hypothesis Testing
Conclusion
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