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CHAPTER 23 PART 3
Examples
From a previous example, a sample of size 20 from insurance companies showed an average
life expectancy 78.6 years and a standard deviation of 4.48 years. We wish to find the sample
size of size n that would allow a 95% confidence level for the mean life expectancy of a policy
holder from the insurance company to have a margin of error of only one year.
We can use 𝑀𝐸 =
∗
𝑡𝑛−1
∙
𝑠
𝑛
and solve for n, but there’s a problem.
Since we don’t yet know n, we don’t
know which critical value t to use.
We can use a critical value z to estimate n, and this
will help us narrow down to an appropriate t to use.
Solution:
We want a 95% confidence level, so we will estimate using z = 1.96.
We want the margin of error to be 1 year.
The standard deviation of the sample was 4.48.
4.48
1 = 1.96 ∙
𝑛
8.7808
1=
𝑛
𝑛 = 8.7808
𝑛 = 77.1
∗
Use n = 78 at 95% confidence. From the table, 𝑡75 is the closest thing. 𝑡75
= 1.992
𝑀𝐸 =
∗
𝑡𝑛−1
𝑠
∙
𝑛
4.48
1 = 1.992 ∙
𝑛
𝑛 = 79.6
Sample Size: 80
In 1960, census results indicated that the age at which American men first married
had a mean of 23.3 years. It is widely suspected that young people today are
waiting longer to get married. We want to find out if the mean age of first
marriage has increased during the past 40 years.
a) Write appropriate hypotheses.
𝐻0 : 𝜇 ≤ 23.3
𝐻𝐴 : 𝜇 > 23.3
In 1960, census results indicated that the age at which American men first married
had a mean of 23.3 years. It is widely suspected that young people today are
waiting longer to get married. We want to find out if the mean age of first
marriage has increased during the past 40 years.
b) We plan to test our hypotheses by selecting a random sample of 40 men who
married for the first time last year. Do you think the necessary assumptions for
inference are satisfied?
Randomization condition: The 40 men were randomly selected.
10% condition: 40 is less than 10% of all recently married men.
Nearly Normal condition: The population of ages of men at first marriage is
likely to be skewed to the right. It is much more likely that there are men who
marry for the first time at an older age than at an age that is much younger
(before 18). We should check for skewness in the distribution of the sample, but
with a sample of size 40, it should be safe to proceed.
In 1960, census results indicated that the age at which American men first married
had a mean of 23.3 years. It is widely suspected that young people today are
waiting longer to get married. We want to find out if the mean age of first
marriage has increased during the past 40 years.
c) Describe the approximate sampling distribution model for the mean age in such
samples.
If the conditions for inference are satisfied, we can model
the sampling distribution of the mean age of men at first
marriage with : 𝑡39 23.3,
𝑠
40
.
In 1960, census results indicated that the age at which American men first married
had a mean of 23.3 years. It is widely suspected that young people today are
waiting longer to get married. We want to find out if the mean age of first
marriage has increased during the past 40 years.
d) The men in our sample married at an average age of 24.2 years, with a
standard deviation of 5.3 years. What’s the P-value for this result?
e) Explain in context what this P-value means.
Using 𝑡 =
24.2−23.3
5.3/ 40
= 1.07, the P-value is 0.1447.
If the mean age at first marriage is still 23.3 years, there is a
14.5% chance of getting a sample mean of 24.2 years or older
simply from natural sampling variation if the null hypothesis is true.
In 1960, census results indicated that the age at which American men first married
had a mean of 23.3 years. It is widely suspected that young people today are
waiting longer to get married. We want to find out if the mean age of first
marriage has increased during the past 40 years.
f) What’s your conclusion?
Since the P-value of 0.1447 is high, we fail to reject the null
hypothesis. There is no evidence to suggest that the mean age
of men at first marriage has changed from 23.3 years, the
mean in 1960.
Today’s Assignment:
Add to HW: page 554 #9-13, 15, 32