Download Lesson 9.4 Testing the Mean µ Part III Notes

Statistics
Lesson 9.4 Testing the Mean µ Part III Notes
Page 1 of 4
Hypothesis testing using p – value – Basic process
1.
State the null and alternate hypothesis and significance level.
2.
Find the sample statistic for normal distribution.
3.
Compute the p – value using the sample statistic.
4.
Compare the p – value and the significance level. Decide whether to reject or fail to reject
the null hypothesis.
5.
Interpret the results in the context of the application.
Part I: Testing µ when σ is known

In most situations, σ is not known; however if there is a prior or preliminary study shown,
the information can be used to get a realistic and accurate value for σ.

How to test µ when σ is known
Let x be a random variable, n be the sample size, x be the mean and σ be known.
1. State the null and alternate hypothesis and set the significance level.
2. If you can assume normal distribution then any sample size will work. If you can no
assume normal distribution, then use a sample size of at least 30. Find the standardized
x
sample test statistic. z x  
n
3. Use the standard normal distribution to find the p – value.
4. Conclude the test. Reject when:
Fail to reject when:
5. Interpret your conclusion.
Part II: Testing µ when σ is Unknown

How to test µ when σ is unknown
Let x be a random variable, n be the sample size, x be the mean and the sample standard
deviation, s.
1. State the null and alternate hypothesis and set the significance level.
2. If you can assume normal distribution then any sample size will work. If you can no assume
normal distribution, then use a sample size of at least 30. Find the standardized sample test
x
statistic. t  s
with a degree of freedom, d.f., = n – 1.
n
3. Use the standard normal distribution to find the p – value.
4. Conclude the test.
Reject when:
Fail to reject when:
5. Interpret your conclusion.
Statistics
Lesson 9.4 Testing the Mean µ Part III Notes
Page 2 of 4
Part III: Testing µ Using Critical Regions

How to test µ when σ is known (using critical regions)
Let x be a random variable, n be the sample size, x be the mean and σ is known (possibly
from a previous study).
1. State the null and alternate hypothesis and set the significance level. The most popular
choice is α = 0.05 or α = 0.01.
2. If you can assume normal distribution then any sample size will work. If you can no assume
normal distribution, then use a sample size of at least 30. Use the σ, the sample size n, the
value of x from the sample and the µ from the null hypothesis to compute the standardized
sample test statistic.
x
zx 

n
3. Show the critical region and critical values on a graph of the sampling distribution. The level
of significance α and the alternate hypothesis determine the locations of critical regions and
critical values.
4. Conclude the test. If the test statistic z computed in Step 2 is in the critical region, then reject
the null hypothesis. If the test statistic z computed in Step 2 is not in the critical region, then
do not reject the null hypothesis.
5. Interpret your conclusion.

How to conclude tests using the critical region method
1. Compute the sample statistic using an appropriate sampling distribution.
2. Using the same sampling distribution, find the critical values as determined by the level of
significance α and the nature of the test: right-tailed, left-tailed, or two-tailed.
3. Compare the sample test statistic to the critical values.
a. For a right-tailed test,
i. If sample test statistic ≥ critical value, reject Ho.
ii. If sample test statistic < critical value, fail to reject Ho.
b. For a left-tailed test,
i. If sample test statistic ≤ critical value, reject Ho.
ii. If sample test statistic > critical value, fail to reject Ho.
c. For a two-tailed test,
i. If sample test statistic lies beyond critical values, reject Ho.
ii. If sample test statistic lies between critical values, fail to reject Ho.
Statistics
Lesson 9.4 Testing the Mean µ Part III Notes
Page 3 of 4
Example 1: Let x be a random variable representing the number of sunspots observed in a four –
week period. A sample of 40 such periods from Spanish colonial times gave the following data.
12.5 14.1 37.6 48.3 67.3 70.0 43.8 56.5 59.7 24.0 54.0 70.1
177.3 4.4
54.6 104
73.9 53.5 27.4 12.0 28.0 13.0 6.50 134.7
114
72.7 81.2 24.1 20.4 13.3 9.4
25.7 47.8 50.0 45.3 61.0
39.0 12.0 7.2
11.3.
The sample mean is x  47.0 . Previous studies indicate that for this period, σ = 35. It is thought
that for thousands of years, the mean number of sunspots per four-week period was about µ = 41.
Do the data indicate that the mean sunspot activity during the Spanish colonial period was higher
than 41? Use α = 0.05. (Solve using critical values.)
Example 2: A professional employee in a large corporation receives an average of µ = 39.8 emails
per day. Most of these e-mails are from other employees in the company. Because of the large
number of e-mails, employees find themselves distracted and are unable to concentrate when they
return to their tasks. In an effort to reduce distraction caused by such interruptions, one company
established a priority list that all employees were to use before sending an email. One month after
the new priority list was put into place, a random sample of 38 employees showed that they were
receiving an average of 32.3 emails per day. The computer server through which the emails are
routed showed that σ = 18.8. Has the new policy has any effect? Use a 1% level of significance to
test the claim that there has been a change in the average number of emails received per day per
employee. What are the critical values?
Statistics
Lesson 9.4 Testing the Mean µ Part III Notes
Page 4 of 4
Example 3:
Let x be a random variable representing dividend yield of Australian bank stocks. We may assume
that the x has normal distribution with σ = 2.4%. A random sample of 11 Australian bank stocks
has a sample mean of x  9.86% . For the entire Australian stock market, the mean dividend is µ =
8.7%. Do these data indicate that the dividend yield of all Australian bank stocks is higher than
8.7%? Use α = 0.01. Find the critical values.
Assignment: Lesson 9.4. p. 378 #4, 6, 9, 18, 27, 29