6.118 Planning another test to compare consumption.
A test of a hypothesis about the mean
consumption of sugar- sweetened beverages at your university based on a sample
of size n=100. The hypotheses are
While the result was not statistically
significant, it did provide some evidence that the mean was smaller than 286.
Thus, you plan to recruit another sample of students from your university but
this time use a onesided alternative. You were thinking of surveying n=100
students but now wonder if this sample size gives adequate power to detect a
decrease of 15 calories per day to μ=271.
(a) Given α=0.05, for what values of z will you reject
the null hypothesis?
(b) Using σ=155 and μ=286 for what
values of x¯ will you reject H0?
(c) Using σ=155 and μ=271, what is the
probability that x¯ will fall in the region defined in part (b)?
(d) Will a sample size of n=100 give
you adequate power? Or do you need to find ways to increase the power? Explain
(e) Use the Statistical
Power applet to determine the sample size n
gives you power near 0.80
6.124 Stress by occupation.
As part of a study on the impact of job
stress on smoking, researchers used data from the Health and Retirement Study
(HRS) to collect information on 3825 ever-smoker individuals who were 50 to 64
years of age.32 An ever-smoker is someone who was a smoker at some time in his
or her life. The HRS is a biennial survey, thus providing the researchers with 17,043
person-year observations. One of the questions on the survey asked a
participant how much he or she agrees or disagrees with the statement “My job
involves a lot of stress.” The answers were coded as a 1 if a participant “strongly
agreed” and 0 otherwise. The following table summarizes these responses by
(a) Because these responses are binary,
use the formula for the standard deviation of a sample
proportion (page 330) and construct 95%
confidence intervals for each occupation.
(b) Summarize the results. Do there
appear to be certain groups of occupations with similar stress levels?
(c) A friend questions the use of the
standard deviation formula in part (a). Refer back to the
binomial setting. What might your friend be concerned with?
6.128 Effect of sample size on significance.
You are testing the null hypothesis
that μ=0 versus the alternative μ>0 using α=0.05 Assume that σ=14. Suppose
that x¯=4 and n=10 Calculate the test statistic and its P-value. Repeat,
assuming the same value of x¯ but with n=20. Do the same for sample sizes of
30, 40, and 50. Plot the values of the test statistic versus the sample size.
Do the same for the P-values.
Summarize what this demonstration shows
about the effect of the sample size on significance testing.
6.129 Blood phosphorus level in dialysis
Patients with chronic
kidney failure may be treated by dialysis, in which a machine removes toxic wastes
from the blood, a function normally performed by the kidneys. Kidney failure
and dialysis can cause other changes, such as retention of phosphorus, that
must be corrected by changes in diet. A study of the nutrition of dialysis
patients measured the level of phosphorus in the blood of several patients on
six occasions. Here are the data for one patient (in milligrams of phosphorus
per deciliter of blood)
5.4 5.2 4.5
4.9 5.7 6.3