155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Chapter 8
Hypothesis Testing
8­1 Review and Preview
8­2 Basics of Hypothesis Testing
8­3 Testing a Claim about a Proportion
8­4 Testing a Claim About a Mean: σ Known
8­5 Testing a Claim About a Mean: σ Not Known
8­6 Testing a Claim About a Standard Deviation or Variance
Requirements for Testing Claims About σ or σ 2
n = sample size
s = sample standard deviation
s2 = sample variance
σ = claimed value of the population standard deviation
σ 2 = claimed value of the population variance
November 08, 2010
Key Concept
This section introduces methods for testing a claim made about a population standard deviation σ or population variance σ 2. The methods of this section use the chi­square distribution that was first introduced in Section 7­5.
Requirements for Testing Claims About σ or σ 2
1. The sample is a simple random sample.
2. The population has a normal distribution. (This is a much stricter requirement than the requirement of a normal distribution when testing claims about means.)
1
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Caution
Chi­Square Distribution
Test Statistic
P­Values and Critical Values for Chi­Square Distribution
November 08, 2010
The χ 2 test of this section is not robust against a departure from normality, meaning that the test does not work well if the population has a distribution that is far from normal. The condition of a normally distributed population is therefore a much stricter requirement in this section than it was in Sections 8­4 and 8­5.
• Use Table A­4.
• The degrees of freedom = n –1.
Properties of Chi­Square Distribution
• All values of χ 2 are nonnegative, and the distribution is not symmetric (see Figure 8­13, following).
• There is a different distribution for each number of degrees of freedom (see Figure 8­14, following).
• The critical values are found in Table A­4 using n – 1 degrees of freedom.
Properties of Chi­Square Distribution ­ cont
Properties of the Chi­
Square Distribution
Figure 8­13
Chi­Square Distribution for 10 and 20 df
Different distribution for each number of df.
Figure 8­14
2
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Table A­4
Table A­4 is based on cumulative areas from the right (unlike the entries in Table A­2, which are cumulative areas from the left). Critical values are found in Table A­4 by first locating the row corresponding to the appropriate number of degrees of freedom (where df = n –1). Next, the significance level α is used to determine the correct column. The following examples are based on a significance level of α = 0.05, but any other significance level can be used in a similar manner.
Table A­4
Two­tailed test:
Unlike the normal and Student t distributions, the 2 critical values in this χ test will be two different positive values (instead of something like ±1.96 ). Divide a significance level of 0.05 between the left and right tails, so the areas to the right of the two critical values are 0.975 and 0.025, respectively. Locate 0.975 and 0.025 at the top of Table A­4.
November 08, 2010
Table A­4
Right­tailed test:
Because the area to the right of the critical value is 0.05, locate 0.05 at the top of Table A­4.
Left­tailed test:
With a left­tailed area of 0.05, the area to the right of the critical value is 0.95, so locate 0.95 at the top of Table A­4.
Example:
A common goal in business and industry is to improve the quality of goods or services by reducing variation. Quality control engineers want to ensure that a product has an acceptable mean, but they also want to produce items of consistent quality so that there will be few defects. If weights of coins have a specified mean but too much variation, some will have weights that are too low or too high, so that vending machines will not work correctly (unlike the stellar performance that they now provide). 3
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Example:
Consider the simple random sample of the 37 weights of post­1983 pennies listed in Data Set 20 in Appendix B. Those 37 weights have a mean of 2.49910 g and a standard deviation of 0.01648 g. U.S. Mint specifications require that pennies be manufactured so that the mean weight is 2.500 g. A hypothesis test will verify that the sample appears to come from a population with a mean of 2.500 g as required, but use a 0.05 significance level to test the claim that the population of weights has a standard deviation less than the specification of 0.0230 g.
Example:
November 08, 2010
Example:
Requirements are satisfied: simple random sample; and STATDISK generated the histogram and quantile plot ­ sample appears to come from a population having a normal distribution. Example:
Step 6: The test statistic is Step 1: Express claim as σ < 0.0230 g
Step 2: If σ < 0.0230 g is false, then σ > 0.0230 g
Step 3: σ < 0.0230 g does not contain equality so it is the alternative hypothesis; null hypothesis is σ = 0.0230 H0: σ = 0.0230 g
H1: σ < 0.0230 g The critical value from Table A­4 corresponds to 36 degrees of freedom and an “area to the right” of 0.95 (based on the significance level of 0.05 for a left­tailed test). Table A­4 does not include 36 degrees of freedom, but Table A­4 shows that the critical value is between 18.493 and 26.509. (Using technology, the critical value is 23.269.) Step 4: significance level is α = 0.05 Step 5: Claim is about σ so use chi­square.
4
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Example:
November 08, 2010
Example:
Step 7: Because the test statistic is in the critical region, reject the null hypothesis. Click globe on left to see Chi­Square Critical Values (xls) at http://cfcc.edu/faculty/cmoore/IndexExcHtm.htm
Statdisk will give critical value of 23.269.
Recap
In this section we have discussed:
• Tests for claims about standard deviation and variance.
• Test statistic.
• Chi­square distribution.
• Critical values.
There is sufficient evidence to support the claim that the standard deviation of weights is less than 0.0230 g. It appears that the variation is less than 0.0230 g as specified, so the manufacturing process is acceptable.
Finding Test Components. In Exercises 5–8, find the test statistic and critical value(s). Also, use Table A­4 to find limits containing the P­value, then determine whether there is sufficient evidence to support the given alternative hypothesis.
453/6. Supermodel Weights. H1: σ < 29 lb, α = 0.05, n = 8, s = 7.5 lb.
5
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
Finding Test Components. In Exercises 5–8, find the test statistic and critical value(s). Also, use Table A­4 to find limits containing the P­value, then determine whether there is sufficient evidence to support the given alternative hypothesis.
453/8. Precipitation Amounts. H1: σ ≠ 0.25, α = 0.01, n = 26, s = 0.18.
Testing Claims About Variation. In Exercises 9–20, test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P­value method or the traditional method of testing hypotheses unless your instructor indicates otherwise.
453/14. Statistics Test Scores Tests in the author’s statistics classes have scores with a standard deviation equal to 14.1. One of his last classes had 27 test scores with a standard deviation of 9.3. Use a 0.01 significance level to test the claim that this class has less variation than other past classes. Does a lower standard deviation suggest that this last class is doing better?
November 08, 2010
Testing Claims About Variation. In Exercises 9–20, test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P­value method or the traditional method of testing hypotheses unless your instructor indicates otherwise.
453/10. Pulse Rates of Men A simple random sample of 40 men results in a standard deviation of 11.3 beats per minute (based on Data Set 1 in Appendix B). 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 Claims About Variation. In Exercises 9–20, test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P­
value method or the traditional method of testing hypotheses unless your instructor indicates otherwise.
454/17. BMI for Miss America Listed below are body mass indexes (BMI) for recent Miss America winners. Use a 0.01 significance level to test the claim that recent Miss America winners are from a population with a standard deviation of 1.34, which was the standard deviation of BMI for winners from the 1920s and 1930s. Do recent winners appear to have variation that is different from that of the 1920s and 1930s?
19.5 20.3 19.6 20.2 17.8 17.9 19.1 18.8 17.6 16.8
= (10 ­ 1)(1.186)2/(1.34)2 = 7.050
Since 7.050 is less than df = 9, the value 7.050 is in the left half of the graph. P­
value = 2*P(χ2 < 7.050) = 2* χ2cdf(0, 7.05, 9) = 0.7362. Do not reject H0; there is not sufficient evidence to reject the claim that σ = 1.34. No, recent winners do not appear to have BMI variation that is different from that of the 1920's and 1930's.
6
155S8.6_3 Testing a Claim About a Standard Deviation or Variance
November 08, 2010
Testing Claims About Variation. In Exercises 9–20, test the given claim. Assume that a simple random sample is selected from a normally distributed population. Use either the P­
value method or the traditional method of testing hypotheses unless your instructor indicates otherwise.
454/20. Playing Times of Popular Songs Listed below are the playing times (in seconds) of songs that were popular at the time of this writing. (The songs are by Timberlake, Furtado, Daughtry, Stefani, Fergie, Akon, Ludacris, Beyonce, Nickelback, Rihanna, Fray, Lavigne, Pink, Mims, Mumidee, and Omarion.) Use a 0.05 significance level to test the claim that the songs are from a population with a standard deviation less than one minute.
448 242 231 246 246 293 280 227 244 213 262 239 213 258 255 257
Since 12.398 is less than df = 15, the value 12.398 is in the left half of the graph. P­
value = P(χ2 < 12.398) = χ2cdf(0, 12.398, 15) = 0.3513. Do not reject H0; there is not sufficient evidence to conclude that σ < 60. There is not sufficient evidence to support the claim that the songs are from a population with a standard deviation less than one minute.
7