the percentage of ash in coal from region 1 has population mean 15 and populatio
ID: 3155388 • Letter: T
Question
the percentage of ash in coal from region 1 has population mean 15 and population standard deviation 4.30. The percentage of ash in coal from region 2 has a normal distribution with unknown mean, , and unknown standard deviation, . In a random sample of coal from region 2, n = 20,x = 12.12 and s= 3.27. (In an actual application, one would need to compute the sample mean and the sample standard deviation.)
(a) With = .10, test to see if the standard deviation for region 2, i.e., , is less than 4.30, which is the standard deviation for region 1. Define a parameter to be used in determining the hypotheses for this testing situation and state the null and alternative hypotheses to be tested in this situation.
(b) What conclusion would be reached for the test in part (a)? State your conclusion in the terminology of this example; not just reject some hypothesis or do not reject a hypothesis.
(c) With = .10, test to see if the standard deviation for region 2, i.e., , is different from 4.30, which is the standard deviation for region 1. Define a parameter to be used in determining the hypotheses for this testing situation and state the null and alternative hypotheses to be tested in this situation.
(d) What conclusion would be reached for the test in part (c)? State your conclusion in the terminology of this example; not just reject some hypothesis or do not reject a hypothesis.
Explanation / Answer
Part a: answer:
Null hypothesis: H0: The population standard deviation for the percentage of ash in coal is 4.30.
Alternative hypothesis: Ha: The population standard deviation for the percentage of ash in coal is less than 4.30.
H0: = 4.30 versus Ha: < 4.30
The population parameter for this test is taken as the population standard deviation.
Part b: answer:
The test statistic formula for this test is given as below:
Chi-square = (n – 1)S^2 / ^2
By plugging all values in the above formula we get the following results
Chi-Square Test of Variance
Data
Null Hypothesis s^2=
4.3
Level of Significance
0.1
Sample Size
20
Sample Standard Deviation
3.27
Intermediate Calculations
Degrees of Freedom
19
Half Area
0.05
Chi-Square Statistic
47.2477
Lower-Tail Test
Lower Critical Value
11.6509
p-Value
0.9997
Do not reject the null hypothesis
Here, we get the p-value as 0.9997 which is greater than the given level of significance or alpha value 0.1, so we do not reject the null hypothesis that the population standard deviation for the percentage of ash in coal is 4.30.
Part c: answer:
Null hypothesis: H0: The population standard deviation for the percentage of ash in coal is 4.30.
Alternative hypothesis: Ha: The population standard deviation for the percentage of ash in coal is different from 4.30.
H0: = 4.30 versus Ha: 4.30
The population parameter for this test is taken as the population standard deviation.
Part d: answer:
The test statistic formula for this test is given as below:
Chi-square = (n – 1)S^2 / ^2
By plugging all values in the above formula we get the following results
Chi-Square Test of Variance
Data
Null Hypothesis s^2=
4.3
Level of Significance
0.1
Sample Size
20
Sample Standard Deviation
3.27
Intermediate Calculations
Degrees of Freedom
19
Half Area
0.05
Chi-Square Statistic
47.2477
Two-Tail Test
Lower Critical Value
10.1170
Upper Critical Value
30.1435
p-Value
0.0003
Reject the null hypothesis
Here, we get the p-value as 0.0003 which is less than the given level of significance or alpha value 0.1, so we reject the null hypothesis that the population standard deviation for the percentage of ash in coal is 4.30. This means we conclude that the population standard deviation for the percentage of ash in coal is different from 4.30.
Chi-Square Test of Variance
Data
Null Hypothesis s^2=
4.3
Level of Significance
0.1
Sample Size
20
Sample Standard Deviation
3.27
Intermediate Calculations
Degrees of Freedom
19
Half Area
0.05
Chi-Square Statistic
47.2477
Lower-Tail Test
Lower Critical Value
11.6509
p-Value
0.9997
Do not reject the null hypothesis
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