t-distibutions: confidence intervals and hypothesis tests
One sample means with unknown sigma
1. A manufacturer of small appliances employs a market research firm to estimate retail
sales of its products by gathering information from a sample of retail stores. A
random sample of 50 stores in the Midwest sales region were asked to give the
number of the manufacturer’s electric can openers that were sold last month. Here
are the data:
19 19 16 19 25 26 24 63 22 16
13 26 34 10 48 16 20 14 13 24
34 14 25 16 26 25 25 26 11 79
17 25 18 15 13 35 17 15 21 12
19 20 32 19 24 19 17 41 24 27
a. Display the data with a histogram and boxplot. Describe the distribution.
b. The distribution of sales is strongly right-skewed because there are many
smaller stores and a few very large stores. The use of procedures here is
reasonably safe despite this violation of the normality assumption. Why?
t
c. Conduct a test at the 5% level to determine if the mean number of can openers
sold last month from all stores in the Midwest sales region is not 19 can
openers. (The conclusion is on the back of this page).
d. Give a 95% confidence interval for the mean number of can openers sold by all
stores in the region. Interpret the interval. (The answer is on the back of this
page).
2. Portable Phones. A manufacturer claims that a new design for a portable phone has
increased the range to 150 feet, allowing many customers to use the phone throughout
their homes and yards. An independent testing laboratory found that a random
sample of 44 of these phones worked over an average distance of 142 feet, with a
standard deviation of 12 feet. Is there evidence that the manufacturer’s claim is too
high? Test at the 1% level. (Conclusion is on the back of this page).
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