1
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.
Although the histogram is unimodal, it shows strong skewness to the right with
a couple of outliers at the high end.
Boxplot reveals three outliers at the high end.
b. The distribution of sales is strongly right-skewed because there are many
smaller stores and a few very large stores. The use of
t
procedures here is
reasonably safe despite this violation of the normality assumption. Why?
Histogram is unimodal which is desirable. Despite strong skewness (according
to the histogram) and several outliers (according to the boxplot), t-procedures
may be used here because
• SRS “A random sample of 50 stores…”
• Sample size is 50 which is greater than 40.
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.
1. 19:
19:
0
≠
=
µ
µ
a
H
H
2.
05.0
=
α
3.
56.23
=
x
523.12≈
x
S
50
=
n
