Stat 301 – Final Exam Review Solutions
1. Recall from the exam 2 review problems the weights of 30 (fun-size) Mounds candy bars and 20 (fun-size) PayDay candy bars, in grams.
d) State null and alternative hypothesis for comparing the means of these two distributions, both in symbols and in words.
Let <![if !vml]><![endif]> Mounds represent the long-run mean weight for the Mounds manufacturing process and <![if !vml]><![endif]> PayDay the long-run meant weight for the PayDay manufacturing process. We want to know whether the observed difference in these sample means convinces us that there is a genuine differences in the long-run mean weights for these two manufacturing processes.
H0: <![if !vml]><![endif]> Mounds - <![if !vml]><![endif]>PayDay = 0 (no difference in the long-run means)
Ha: <![if !vml]><![endif]> Mounds - <![if !vml]><![endif]> PayDay ≠ 0 (this is a difference in the long-run means)
We were just asked to compare the means, there was no prior suspicion given as to which candy bars would be heavier, so we are using a two-sided alternative.
e) Do you think a theory-based analysis would be appropriate for these data? Explain how you are deciding.
It’s a little questionable, the sample sizes are both at least 20 but we do have some skewness in the sample distributions (and in opposite directions). Would probably be worth looking at a second analysis as well.
d) Is this distribution approximately normal? Would you have expected this? Explain
This distribution looks a bit skewed left, consistent with our skepticism in question (c).
e) Would you expect this distribution to follow a t distribution? Explain
Actually, the t model is for the standardized statistics, not the distribution of the differences in sample means. We don’t quite see the heaviness in the tails (because we aren’t using the s’s yet) that we might for the t distribution.
f) Use the above output to roughly approximate the p-value. Explain how.
First we have to roughly approximate the difference in sample means from the original dotplots, keeping in mind that the PayDay mean might be pulled a little bit to the left of the main peak. Maybe 22.2 grams and 20.5 grams?? So say we think the difference in means is around 2 grams. Then we need to look at the randomization distribution and see how often we get a difference of -2 or smaller or a difference of 2 or larger for a two-sided p-value. However, we see 2 is off the chart on both ends so this would approximate a p-value of zero. In reality the difference in means is 1.605 grams, which is still pretty extreme and the empirical p-value is zero.
g) Explain a difficulty with using this simulation approach to analyze these data.
This is a randomization distribution, assuming random shuffling of the weights to the two brands. But that is not how the data were collected, that was through independent random sampling from each random process. So we might prefer a simulation that reflecting the randomness from sampling from an infinite process rather than from random assignment (e.g., bootstrapping).
h) Assuming it’s valid, how would you interpret this confidence interval?
I’m 95% confident that the mean weight of (all) PayDay candy bars is .99 to 2.2 gram larger than the mean weight of (all) Mounds candy bars.
2) A study examined whether a nicotine lozenge can help a smoker to quit. The research reports on many background variables, such as age, weight, gender, number of cigarettes smoked, and whether the person made a previous attempt to quit smoking (Shiffman et al., 2002). Suppose the researchers want to compare the distributions of the background variables between the two treatment groups (nicotine lozenge or placebo lozenge).
(a) For each of the five variables listed, indicate whether it calls for a comparison of means or a comparison of proportions.
Age – quantitative – means
Weight – quantitative – means
Gender – categorical – proportions
Number of cigs smoked – quantitative - means
Previous attempt – categorical – proportions
(b) Would the researchers hope to reject the null hypotheses or fail to reject the null hypotheses in these tests? Explain.
In this case, large p-values would be good news – it would provide more evidence that our treatment groups were similar to each other at the beginning of the study so any differences observed at the end of the study are even more safely attributed to the nicotine lozenge’s superior effectiveness to the placebo lozenge.
(c) Of the 459 nicotine users, 46.0% successfully abstained (didn’t start smoking again) for 6 weeks, compared to 29.7% of the 458 control group (without nicotine). Calculate and interpret a 95% confidence interval.
<![if !vml]><![endif]><![if !vml]><![endif]>
I’m 95% confident that the probability of abstaining is .1009 to .2246 larger when assigned to nicotine rather than to the placebo.
(d) Are you willing to draw a cause-and-effect conclusion from this study? If not, suggest a possible confounding variable and explain how it is confounding in this study.
Yes, because there was random assignment to the treatment groups (nicotine or not)
(e) Are you willing to generalize these results to all smokers interested in quitting? If not, suggest a possible source of sampling bias and the likely direction of the bias.
Maybe not, we don’t have a lot of information about these individuals were recruited. Maybe those willing to participate in a smoking cessation study are different (more likely to abstain) than those who want to quite but aren’t willing to participate in a research study.
3) Researchers examined the long-term survival of doctors graduating from one medical school over one century (Redelmeier and Kwong, 2004), comparing those who were presidents of their class to those who appeared alphabetically before or alphabetically after the president in the graduating class photograph. Statistics on long-term mortality were obtained from licensing authorities, medical obituaries, professional associations, alumni records, and national physician directories (follow-up 94%). They reported on 507 presidents and 1014 classmates.
(a) Is it reasonable to treat the presidents and non-presidents as independent random samples?
This is a bit debatable, because they took all three from the same year, rather than taking a random sample of presidents and then a separate random sample of non presidents. But if we don’t think the variable changes too much from year to year, we could treat them as independent samples. It might be better to more directly compare the 3 classmates from each graduating class, but it’s not obvious how to do that either. It’s not clear how the 507 were selected or is that all of the presidents. This is also from just one school.
Assuming the answer to (a) is yes:
(b) The researchers examined several base-line variables, including gender and whether or not the individual wore glasses. They found 93% of the presidents were male, compared to 85% of their classmates. They also found 9% of presidents were glasses, compare to 12% of their classmates. Are either of these differences statistically significant?
Because the response variables here (sex and whether wore glasses) are categorical, we will consider the two-sample z procedure. In both cases, we can define <![if !vml]><![endif]>pres <![if !vml]><![endif]>classmate as the parameter of interest and then test hypotheses H0: <![if !vml]><![endif]>pres-<![if !vml]><![endif]>classmate = 0 (no difference in the population proportions) vs. Ha: <![if !vml]><![endif]>pres-<![if !vml]><![endif]>classmate ≠ 0 (there is a difference in the population proportions). Because the sample sizes are large, the two-sample z-procedures are appropriate.
JMP output for male/female comparison:
There is a statistically significant difference (p-value <.001 < .05) in the sample proportion of presidents who were male compared to the sample proportion of classmates who were male. We are 95% confident that the population proportion of classmates who are male is .05 to .11 lower than the population proportion of presidents who are male.
Applet and JMP output for glasses comparison:
<![if !vml]><![endif]><![if !vml]><![endif]>
There is not a statistically significant difference (p-value = .1187 > 0.05) in the proportion wearing glasses.
(c) The overall-life expectancy for the presidents was 49.0 years compared to 51.4 years for their classmates. The two-sided p-value was reported to be .036. Assuming the sample standard deviations were similar in the two samples, use trial-and-error in JMP or TBI applet or algebra to approximate the value of this standard deviation. What conclusion would you draw from this p-value?
Let <![if !vml]><![endif]>pres represent the life-expectancy (mean lifetime) for all class presidents and <![if !vml]><![endif]>class represent the life-expectancy for all classmates.
H0: <![if !vml]><![endif]>pres - mclass = 0 (no difference in the average life-expectancy between these two populations)
Ha: <![if !vml]><![endif]>pres ≠ <![if !vml]><![endif]>class (there is a difference)
If we use the conservative df of 506, a two-sided p-value of .036 corresponds to t0 = -2.102.
<![if !vml]><![endif]><![if !vml]><![endif]> =439.4, so s = 20.96.
The sample standard deviation must have been close to 21 years.
Using this value below to verify the p-value:
The p-value of .036 provides moderate evidence (.01 < p-value < .05) of a difference between the population mean life expectancy of the class presidents compared to their classmates. We would reject the null hypothesis at the 5% level.
4) Because these were two different questions on the same survey, each respondent is answering both questions, we shouldn’t apply a “two-sample” procedure, but should treat the observations as paired instead.
5) In a study reported in the July 6, 2007 issue of the journal Science, researchers studied 396 American college students and kept track of each student’s sex and also how many words they spoke in a day. They found that females spoke an average of 16,215 words per day and males an average of 15,669 words per day.
Consider the following variables:
For each research question below, which theory-based method would you consider:
<![if !supportLists]>· <![endif]>One-proportion z-test or interval
<![if !supportLists]>· <![endif]>One-mean t-test or interval
<![if !supportLists]>· <![endif]>Two-proportion z-test or interval
<![if !supportLists]>· <![endif]>Two-mean t-test or interval
Briefly justify your answer.
(a) Do women tend to use more words than men?
(b) How often does the proportion of adjectives a person uses in a day exceed 0.25? In other words, estimate the probability more than 25% of the words someone uses in a day are adjectives.
Confidence interval for one proportion (variable: whether or not exceed .25; parameter: probability of exceeded 0.25.)
(c) Are women more likely than men to use more than 15,000 words per day?
(d) Do people tend to talk more (use more words) on the weekends or on the weekdays?
Paired t-test (difference in number of words) or one proportion (probability of using more on weekend than weekday). The point is get one measurement per person
6)The Roller Coaster Database maintains a web site (www.rcdb.com) with data on roller coasters around the world. Some of the data recorded include whether the coaster is made of wood or steel and the maximum speed achieved by the coaster, in miles per hour. The boxplots display the distributions of speed by type of coaster for 145 coasters in the United States as of Nov. 2003.
(a) Do these boxplots allow you to determine whether there are more wooden or steel roller coasters?
No, no sample size information presented
(b) Do these boxplots allow you to say which type has a higher percentage of coasters that go faster than 60mph? Explain and, if so, answer the question.
50% of steel go faster than 60 mph compared to 25% of wooden
(c) Do these boxplots allow you to say which type has a higher percentage of coasters that go faster than 50mph? Explain and, if so, answer the question.
Both types have 75% exceeding 50 mph.
(d) Do these boxplots allow you to say which type has a higher percentage of coasters that go faster than 48mph? Explain and, if so, answer the question.
No, because 48 mph does not match up with a quartile, we can’t say anything about how the values compare to 48mph. We know nothing about how the lowest 25% are distributed along those “whiskers.” In particular, having longer whiskers doesn’t imply a higher percentage in that area.
(e) The steel coasters have a “high outlier.” Explain how I know this from the above display and interpret this outlier in context. What would be your next step in analyzing these data?
The star plotted off on its own. This is a coaster than goes much faster than all the rest. We should figure out which coaster it is. We can also see how the distribution changes if that observation is removed.
(f) Conjecture as to how the mean, median, interquartile range, and standard deviation will change (if at all) if that coaster identified in part (e) (Top Thrill Dragster in Cedar Point Amusement Park, Sandusky, Ohio) is removed from the data set. Explain your reasoning.
Removing a high outlier will lower the mean and even the median, but probably more noticeably for the mean.
Removing a high outlier that is far from the mean will lower the variability in the data so the interquartile range and standard deviation will decrease, but probably more noticeably for the standard deviation.