3. Use the following SAT score data to answer the questions that follow.
Gender Verbal Math
M 600 690
M 610 550
F 490 800
F 680 610
M 520 540
F 680 660
M 650 700
M 600 560
F 550 560
M 490 390
F 530 530
M 560 560
F 630 590
F 510 520
M 710 740
F 550 560
M 690 620
M 700 700
M 540 620
F 280 500
M 710 760
F 640 710
M 600 590
M 610 670
M 680 670
a. Use this sample to compute a 90% confidence interval for the mean SAT Math score at this school.
b. The national mean Math score is 503. Use your confidence interval and compare the performance of these students to the national mean. State your conclusions.
c. Set up and test an appropriate hypothesis to determine if the mean Math score at this school differs from the national mean of 503. Please show all the parts of your test: i.e. the null hypothesis, the alternative hypothesis, either the z-score or the p-value, and your conclusion stated mathematically and as a sentence in the context of the problem. Use alpha = .05.
d. Use this sample to compute a 90% confidence interval for the mean SAT Verbal score for Females at this school.
e. Use this sample to compute a 90% confidence interval for the mean SAT Verbal score for Males at this school.
im not asking fo the answers i just need help getting started. thank you
To answer these questions, we will perform the following steps:
a. Compute a 90% confidence interval for the mean SAT Math score at this school:
1. Calculate the sample mean (X̄) and sample standard deviation (s) of the SAT Math scores.
2. Use the formula for confidence interval: CI = X̄ ± (Z * (s / √n)), where Z is the critical value for the desired confidence level and n is the sample size.
3. Look up the appropriate critical value for a 90% confidence level in the Z-table or use software.
4. Substitute the values into the formula to calculate the lower and upper bounds of the confidence interval.
b. Compare the performance of these students to the national mean Math score of 503:
1. Compare the lower and upper bounds of the confidence interval obtained in part (a) with the national mean Math score of 503.
2. If the confidence interval contains the national mean (503), conclude that the scores of these students are not significantly different from the national mean. Otherwise, if the confidence interval does not contain the national mean, conclude that the scores of these students are significantly different from the national mean.
c. Set up and test an appropriate hypothesis to determine if the mean Math score at this school differs from the national mean of 503:
1. Null Hypothesis (H0): The mean Math score at this school is equal to the national mean (µ = 503).
2. Alternative Hypothesis (Ha): The mean Math score at this school is not equal to the national mean (µ ≠ 503).
3. Calculate the test statistic: Z = (X̄ - µ) / (σ / √n), where X̄ is the sample mean, µ is the population mean, σ is the population standard deviation (not given), and n is the sample size.
4. Look up the Z-score in the Z-table or use software to find the p-value associated with the test statistic.
5. Compare the p-value with the significance level (alpha = 0.05).
6. If the p-value is less than alpha, reject the null hypothesis and conclude that the mean Math score at this school differs from the national mean. Otherwise, if the p-value is greater than alpha, fail to reject the null hypothesis and conclude that there is not enough evidence to suggest a difference.
d. Compute a 90% confidence interval for the mean SAT Verbal score for Females at this school:
1. Extract the SAT Verbal scores for only the female students from the data.
2. Calculate the sample mean (X̄) and sample standard deviation (s) of the SAT Verbal scores for females.
3. Use the formula for confidence interval: CI = X̄ ± (Z * (s / √n)), where Z is the critical value for the desired confidence level and n is the sample size.
4. Look up the appropriate critical value for a 90% confidence level in the Z-table or use software.
5. Substitute the values into the formula to calculate the lower and upper bounds of the confidence interval.
e. Compute a 90% confidence interval for the mean SAT Verbal score for Males at this school:
1. Extract the SAT Verbal scores for only the male students from the data.
2. Calculate the sample mean (X̄) and sample standard deviation (s) of the SAT Verbal scores for males.
3. Use the formula for confidence interval: CI = X̄ ± (Z * (s / √n)), where Z is the critical value for the desired confidence level and n is the sample size.
4. Look up the appropriate critical value for a 90% confidence level in the Z-table or use software.
5. Substitute the values into the formula to calculate the lower and upper bounds of the confidence interval.