Question 1: The following data set represents
the repair costs (in dollars) for a random sample of 30 dishwashers
|
41.82
|
52.81
|
57.8
|
68.16
|
73.48
|
78.88
|
88.13
|
88.79
|
|
90.07
|
90.35
|
91.68
|
91.72
|
93.01
|
95.21
|
95.34
|
96.5
|
|
100.05
|
101.32
|
103.59
|
104.19
|
105.62
|
111.32
|
117.14
|
118.42
|
|
118.77
|
119.01
|
120.7
|
140.52
|
141.84
|
147.06
|
|
|
(a) Find the point estimate of the population mean.
(b) Find the margin of error for a 95% level of confidence.
(c) Construct a 95% confidence interval for the population
mean and interpret the results.
Solution:
(a) The point estimate of the population mean is
computed as:
(b) By definition, the margin of error is computed
as:
where corresponds to the cutoff point of
t-distribution, for ,
and 31 degrees of freedom, which means that
Also, using Excel we find that
This means that:
(c) The 95% confidence interval is computed as:
This means that there's a 95% chance that the interval
(88.90429, 107.3157) contains the actual population mean .
Question 2: In Illinois,
a random sample of 85 eighth grade students has a mean score of 265 with a
standard deviation of 55 on a national mathematics assessment test. This test
result prompts a state school administrator to declare that the mean score for
that state's eighth graders on the examination is more than 260. At ,
is there enough evidence to support the administrator's claim?
Solution:
(a) The null and alternative hypotheses are:
This corresponds to a right-tailed test. The population
variance is not known, but the sample size is big enough to use a Z-test.
(b) The Z-statistics is computed as:
(c) The p-value is computed as:
(d) Since the p-value found is greater than 0.04, we
fail to reject the null hypothesis.
(e) We don't have enough evidence to support the
claim that the mean score is greater than 260, at the 0.04 significance
level.
