Question

Suppose the average Math SAT score for all students taking the exam this year is 480 with standard deviation 100. Assume the distribution of scores is normal. A SRS of four students is selected and given special training to prepare for the Math SAT. The mean Math SAT score of these students is found to be 560, 80 points higher than the national average. We may correctly concludeA. the results are neither statistically significant at level = 0.05 nor practically significant.

B. the results are not statistically significant at level = 0.05, but they are practically significant.
C. the results are statistically significant at level = 0.05, but they are not practically significant.

Answer 1

The results are statistically significant at level = 0.05 and they are also practically significant.

Step-by-step explanation:

To determine if the results are statistically significant, we need to perform a hypothesis test. Let's define the null hypothesis as the mean Math SAT score for the special training group is equal to the average Math SAT score for all students, and the alternative hypothesis as the mean Math SAT score for the special training group is greater than the average Math SAT score for all students.

Next, we calculate the test statistic, which is the z-score. The formula to calculate the z-score is: z = (x - mu) / (sigma / sqrt(n)), where x is the sample mean, mu is the population mean, sigma is the population standard deviation, and n is the sample size. Plugging in the values, we get: z = (560 - 480) / (100 / sqrt(4)) = 8.

Finally, we compare the z-score to the critical value of the standard normal distribution at a significance level of 0.05. The critical value for a one-tailed test is 1.645. Since the test statistic of 8 is greater than the critical value of 1.645, we reject the null hypothesis. Therefore, we can conclude that the results are statistically significant at level = 0.05.

Additionally, since the mean Math SAT score for the special training group is 80 points higher than the national average, we can also say that the results are practically significant, as there is a significant improvement in scores after the special training.

Answer 2

B. the results are not statistically significant at level = 0.05, but they are practically significant.

Step-by-step explanation:

Given that the average Math SAT score for all students taking the exam this year is 480 with standard deviation 100.

Sample size n =4

`\bar x =550\\`

mean difference = 550-480 = 70

Std error of sample =
`(100)/(√(4) ) =50`

Since sample size is small t test can be done with df =3

`H_0: \bar x = 480\\H_a: \bar x >480`

(one tailed test)

t =
`(70)/(50) =1.4`

p value =0.128

Since p >0.05 we accept null hypothesis.

B. the results are not statistically significant at level = 0.05, but they are practically significant.