4.2 Identify the parameter, Part II. For each of the following situations, state whether the parameter of interest is a mean or a proportion.
(a) A poll shows that 64% of Americans personally worry a great deal about federal spending and the budget deficit.
(b) A survey reports that local TV news has shown a 17% increase in revenue between 2009 and 2011 while newspaper revenues decreased by 6.4% during this time period.
(c) In a survey, high school and college students are asked whether or not they use geolocation services on their smart phones.
(d) In a survey, internet users are asked whether or not they purchased any Groupon coupons.
(e) In a survey, internet users are asked how many Groupon coupons they purchased over the last year.
4.4 Heights of adults. Researchers studying anthropometry collected body girth measurements and skeletal diameter measurements, as well as age, weight, height and gender, for 507 physically active individuals. The histogram below shows the sample distribution of heights in centimeters.
(a) What is the point estimate for the average height of active individuals? What about the median?
(b) What is the point estimate for the standard deviation of the heights of active individuals?
What about the IQR?
(c) Is a person who is 1m 80cm (180 cm) tall considered unusually tall? And is a person who is 1m 55cm (155cm) considered unusually short? Explain your reasoning.
(d) The researchers take another random sample of physically active individuals. Would you
expect the mean and the standard deviation of this new sample to be the ones given above.
Explain your reasoning.
(e) The samples means obtained are point estimates for the mean height of all active individuals, if the sample of individuals is equivalent to a simple random sample. What measure do we use to quantify the variability of such an estimate? Compute this quantity using the data from the original sample under the condition that the data are a simple random sample.
4.6 Chocolate chip cookies. Students are asked to count the number of chocolate chips in 22 cookies for a class activity. They found that the cookies on average had 14.77 chocolate chips with a standard deviation of 4.37 chocolate chips.
(a) Based on this information, about how much variability should they expect to see in the mean number of chocolate chips in random samples of 22 chocolate chip cookies?
(b) The packaging for these cookies claims that there are at least 20 chocolate chips per cookie. One student thinks this number is unreasonably high since the average they found is much lower. Another student claims the di_erence might be due to chance. What do you think?
4.8 Mental health. Another question on the General Social Survey introduced in Exercise 4.7 is \For how many days during the past 30 days was your mental health, which includes stress, depression, and problems with emotions, not good?" Based on responses from 1,151 US residents, the survey reported a 95% confidence interval of 3.40 to 4.24 days in 2010.
(a) Interpret this interval in context of the data.
(b) What does a 95% confidence level mean in this context?
(c) Suppose the researchers think a 99% confidence level would be more appropriate for this interval. Will this new interval be smaller or larger than the 95% confidence interval?
(d) If a new survey asking the same questions was to be done with 500 Americans, would the standard error of the estimate be larger, smaller, or about the same. Assume the standard deviation has remained constant since 2010.
4.10 Confidence levels. If a higher con_dence level means that we are more confident about the number we are reporting, why don't we always report a confidence interval with the highest possible confidence level?
4.12 Thanksgiving spending, Part I. The 2009 holiday retail season, which kicked off on November 27, 2009 (the day after Thanksgiving), had been marked by somewhat lower self-reported consumer spending than was seen during the comparable period in 2008. To get an estimate of consumer spending, 436 randomly sampled American adults were surveyed. Daily consumer spending for the six-day period after Thanksgiving, spanning the Black Friday weekend and Cyber Monday, averaged $84.71. A 95% confidence interval based on this sample is ($80.31, $89.11). Determine whether the following statements are true or false, and explain your reasoning.
(a) We are 95% confident that the average spending of these 436 American adults is between $80.31 and $89.11.
(b) This confidence interval is not valid since the distribution of spending in the sample is right skewed.
(c) 95% of such random samples would have a sample mean between $80.31 and $89.11.
(d) We are 95% confident that the average spending of all American adults is between $80.31 and $89.11.
(e) A 90% confidence interval would be narrower than the 95% confidence interval since we don't need to be as sure about capturing the parameter.
(f) In order to decrease the margin of error of a 95% confidence interval to a third of what it is now, we would need to use a sample 3 times larger.
(g) The margin of error for the reported interval is 4.4.