Transcript Hypothesis Testing for Standard Deviation or Variance

Hypothesis Testing for Standard
Deviation or Variance
Testing a Claim about a Standard Deviation or
Variance
We first need to make sure we meet the requirements.
1. The sample observations are a simple random sample.
2. The population has a normal distribution.
Test Statistic for Testing a Claim about a 𝛔 or 𝝈𝟐 .
2
(𝑛
−
1)𝑠
𝜒2 =
𝜎2
Testing a Claim about 𝜎 or 𝜎 2
Critical-value method in 5 Steps
1. State the hypothesis and state the claim.
2. Compute the test value. (Involves find the sample
statistic).
3. Draw a picture and find Critical Values.
4. Make the decision to reject 𝐻0 or not. (compare P-value
and 𝛼)
5. Summarize the results.
Testing a Claim about 𝜎 or 𝜎 2
When 40 people used the Weight Watchers diet for one year,
their weights losses had a standard deviation of 4.9lb. Use a
0.01 significance level to test the claim that the amounts of
weight loss have a standard deviation equal to 6.0lb, which
appears to be the standard deviation for the amounts of
weight loss with the zone diet.
1. 𝐻0 : 𝜇 = 6.0𝑙𝑏 claim and 𝐻𝑎 : 𝜇 ≠ 6.0𝑙𝑏
2.
(𝑛−1)𝑠 2
(40−1)4.92
=
=
= 26.011
𝜎2
6.02
Critical 𝜒 2 value= 20.707 and 66.766
𝜒2
3.
4. Fail to reject 𝐻0 since our test stat 26.011 is not in the critical
region.
5. There is not sufficient evidence to warrant rejection of the
claim.
Testing a Claim about 𝜎 or 𝜎 2
A simple random sample of 40 men results in a standard
deviation of 11.3 beats per minute. The normal range of pulse
rates of adults is typically given as 60 to 100 beats per minute. If
the range rule of thumb is applied to that normal range, the
result is a standard deviation of 10 beats per minute. Use the
sample results with a 0.05 significance level to test the claim that
pulse rates of men have a standard deviation greater than 10
beats per minute.
Testing a Claim about 𝜎 or 𝜎 2
Listed below are the playing times of songs that were popular at
the time of this writing. Use a 0.05 significance level to test the
claim that the songs are from a population with standard
deviation less than one minute.
448 242 231 246 246 293 280 227 244 213 262 239 213
258 255 257