MATHEMATICS

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.

4.14 Age at first marriage, Part I. The National Survey of Family Growth conducted by the

Centers for Disease Control gathers information on family life, marriage and divorce, pregnancy,

infertility, use of contraception, and men’s and women’s health. One of the variables collected on

this survey is the age at first marriage. The histogram below shows the distribution of ages at

first marriage of 5,534 randomly sampled women between 2006 and 2010. The average age at first

marriage among these women is 23.44 with a standard deviation of 4.72

Estimate the average age at _rst marriage of women using a 95% confidence interval, and interpret

this interval in context. Discuss any relevant assumptions.

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