Assume that Science scores on a 25-point quiz for a random sample of 5 students were drawn from a
normal population. The assumed variance was 20 while the recorded variances were: 18, 16, 10, 13, and
23. You suspected that the variance is actually higher.
What test do you need to use?
State the null and alternative hypotheses.
Test the hypotheses at the 10% significance level. What is the p-value?
What is the conclusion (2-part)?
Estimate the population variance with 90% confidence.
To test whether the variance is actually higher than the assumed variance, you need to use a one-tailed F-test. This test compares the ratio of two sample variances to determine if they are significantly different.
Null hypothesis (H0): The population variance is equal to the assumed variance.
Alternative hypothesis (Ha): The population variance is greater than the assumed variance.
To test the hypotheses at the 10% significance level, you will calculate the p-value. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. If the p-value is less than the significance level (10% in this case), you reject the null hypothesis.
To calculate the p-value, you need to find the F-statistic and compare it to the critical value for the F-distribution.
Step 1: Calculate the sample variance of the recorded variances:
- Sample variance = (18 + 16 + 10 + 13 + 23) / 5 = 16.
Step 2: Calculate the test statistic (F-statistic):
- F = (Sample variance) / (Assumed variance) = 16 / 20 = 0.8.
Step 3: Calculate the degrees of freedom for the F-distribution:
- Degrees of freedom numerator = Number of recorded variances - 1 = 5 - 1 = 4.
- Degrees of freedom denominator = Number of students in the sample - 1 = 5 - 1 = 4.
Step 4: Look up the critical value in the F-distribution table at the given significance level and degrees of freedom. In this case, because it's a one-tailed test, you'll look for the critical value on the right side of the F-distribution.
Step 5: Determine the p-value by comparing the F-statistic to the critical value. If the F-statistic is greater than the critical value, the p-value is less than the significance level, and you reject the null hypothesis.
The conclusion will depend on the values obtained from the F-distribution table or a statistical software. Without these values, it's not possible to provide an exact conclusion or calculate the p-value.
To estimate the population variance with 90% confidence, you can use the confidence interval formula for the variance. The formula is given by:
Confidence Interval for Variance = [(n - 1) * Sample variance] / Chi-square upper critical value, (α/2), DF
Here, n is the sample size, Sample variance is the calculated sample variance, Chi-square upper critical value is obtained from the Chi-square distribution table for the desired confidence level (90% in this case), and DF is the degrees of freedom.
To calculate the confidence interval, you will need the degrees of freedom and the Chi-square upper critical value for the given confidence level. Without these specific values, it's not possible to provide an exact estimate of the population variance with 90% confidence.