Answer :
To test the administrator's claim that the standard deviation of SAT critical reading test scores is 112, we can perform a chi-square test for a single sample variance. Here's how you would perform the test step by step:
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H₀): The population standard deviation is equal to the claimed value, which is σ = 112.
- Alternative Hypothesis (H₁): The population standard deviation is not equal to the claimed value, which is σ ≠ 112.
Step 2: Set the level of significance.
- We are given a significance level of α = 0.10.
Step 3: Calculate the test statistic.
- For a chi-square test for variance, the test statistic is calculated using the formula:
[tex]\[ \chi^2 = \frac{(n - 1)s^2}{\sigma^2} \][/tex]
Where:
n is the sample size
s is the sample standard deviation
σ is the claimed population standard deviation
- Plugging the values we have:
[tex]\[ \chi^2 = \frac{(19 - 1) 143^2}{112^2} \][/tex]
[tex]\[ \chi^2 = \frac{18 20449}{12544} \][/tex]
[tex]\[ \chi^2 = \frac{368082}{12544} \][/tex]
[tex]\[ \chi^2 \approx 29.35 \][/tex]
Step 4: Determine the critical value.
- The critical value for a two-tailed chi-square test can be found using a chi-square distribution table or calculator, with degree of freedom df = n - 1 and the level of significance α. You would usually look up the value in a chi-square distribution table or use a statistical software or a calculator.
- For our example, degrees of freedom (df) = 19 - 1 = 18.
- We are looking at a two-tailed test because we are testing for a difference in either direction (whether the actual standard deviation is greater or less than the claimed standard deviation).
- The critical values are the points where if the calculated chi-square value falls beyond them, we reject H₀. These will correspond to the upper and lower percentage points of the chi-square distribution. The lower critical value will be at the α/2 percentile, and the upper critical value will be at the (1 - α/2) percentile.
Step 5: Make a decision.
- If the computed chi-square test statistic is greater than the upper critical value or less than the lower critical value, we reject the null hypothesis.
- If our calculated chi-square value falls between the two critical values, we fail to reject the null hypothesis.
Step 6: Conclusion.
- Depending on the critical value obtained and comparing it with our chi-square value of approximately 29.35, we make our decision on whether to reject or fail to reject the null hypothesis.
- If the test statistic (29.35) is greater than the upper critical chi-square value or less than the lower critical chi-square value at 18 degrees of freedom and α = 0.10, we reject the null hypothesis, suggesting that there is enough evidence at the 0.10 significance level to reject the administrator's claim that the standard deviation of SAT critical reading test scores is 112.
- If the test statistic falls within the critical value range, we fail to reject the null hypothesis, suggesting that there is not enough evidence at the 0.10 significance level to reject the administrator's claim.
Step 1: State the null and alternative hypotheses.
- Null Hypothesis (H₀): The population standard deviation is equal to the claimed value, which is σ = 112.
- Alternative Hypothesis (H₁): The population standard deviation is not equal to the claimed value, which is σ ≠ 112.
Step 2: Set the level of significance.
- We are given a significance level of α = 0.10.
Step 3: Calculate the test statistic.
- For a chi-square test for variance, the test statistic is calculated using the formula:
[tex]\[ \chi^2 = \frac{(n - 1)s^2}{\sigma^2} \][/tex]
Where:
n is the sample size
s is the sample standard deviation
σ is the claimed population standard deviation
- Plugging the values we have:
[tex]\[ \chi^2 = \frac{(19 - 1) 143^2}{112^2} \][/tex]
[tex]\[ \chi^2 = \frac{18 20449}{12544} \][/tex]
[tex]\[ \chi^2 = \frac{368082}{12544} \][/tex]
[tex]\[ \chi^2 \approx 29.35 \][/tex]
Step 4: Determine the critical value.
- The critical value for a two-tailed chi-square test can be found using a chi-square distribution table or calculator, with degree of freedom df = n - 1 and the level of significance α. You would usually look up the value in a chi-square distribution table or use a statistical software or a calculator.
- For our example, degrees of freedom (df) = 19 - 1 = 18.
- We are looking at a two-tailed test because we are testing for a difference in either direction (whether the actual standard deviation is greater or less than the claimed standard deviation).
- The critical values are the points where if the calculated chi-square value falls beyond them, we reject H₀. These will correspond to the upper and lower percentage points of the chi-square distribution. The lower critical value will be at the α/2 percentile, and the upper critical value will be at the (1 - α/2) percentile.
Step 5: Make a decision.
- If the computed chi-square test statistic is greater than the upper critical value or less than the lower critical value, we reject the null hypothesis.
- If our calculated chi-square value falls between the two critical values, we fail to reject the null hypothesis.
Step 6: Conclusion.
- Depending on the critical value obtained and comparing it with our chi-square value of approximately 29.35, we make our decision on whether to reject or fail to reject the null hypothesis.
- If the test statistic (29.35) is greater than the upper critical chi-square value or less than the lower critical chi-square value at 18 degrees of freedom and α = 0.10, we reject the null hypothesis, suggesting that there is enough evidence at the 0.10 significance level to reject the administrator's claim that the standard deviation of SAT critical reading test scores is 112.
- If the test statistic falls within the critical value range, we fail to reject the null hypothesis, suggesting that there is not enough evidence at the 0.10 significance level to reject the administrator's claim.
