SAT Exam Scores – A school administrator wonders if students whose first language is not English score differently on the math portion of the SAT exam than students whose first language is English. The mean SAT math score of students whose first language is English is 516, on the basis of data obtained from the College Board. A simple random sample of 20 students whose first language is not English results in a sample mean SAT math score of 522. SAT math scores are normally distributed, with a population standard deviation of 114.

(a) Why is it necessary for SAT math scores to be normally distributed to test the hypotheses using the methods of this section?
(b) Determine the appropriate null and alternative hypotheses to assess whether students whose first language is not English score differently on the math portion of the SAT exam.
(c) Use the classical or P-value approach at the á = 0.1 level of significance to test the hypothesis in part (b).
(d) Write a conclusion based on your results to part (c)

a. What methods were in the section? Probabilities can be calculated for a normal distribution.

b. Ho: mean 1 = mean 2

Ha: mean 1 ≠ mean 2

c. Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√(n-1)

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score.

d. Is that proportion from c ≤ .01 (.005 at each end)?

if-x=76.9s=8.5compute the test statistic

(a) It is necessary for SAT math scores to be normally distributed in order to use the methods of hypothesis testing because these methods are based on assumptions of normality. When the scores are normally distributed, it allows us to make accurate inferences and draw conclusions using the appropriate statistical tests.

(b) The appropriate null and alternative hypotheses to assess whether students whose first language is not English score differently on the math portion of the SAT exam are as follows:
Null hypothesis (H0): The mean SAT math score for students whose first language is not English is equal to the mean SAT math score for students whose first language is English.
Alternative hypothesis (Ha): The mean SAT math score for students whose first language is not English is different from the mean SAT math score for students whose first language is English.

(c) To test the hypothesis in part (b), we can use the classical or P-value approach. In this case, we will use the classical approach.

Using the classical approach, we first need to compute the test statistic. The formula for the test statistic in this case is:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

Given:
Sample mean (x̄) = 522
Population mean (µ) = 516
Population standard deviation (σ) = 114
Sample size (n) = 20

Substituting the values into the formula, we have:
z = (522 - 516) / (114 / sqrt(20))

= 6 / (114 / 4.472)

= 6 / 25.343

= 0.237

Next, we need to compare the test statistic to the critical value. At α = 0.1 level of significance, the critical value for a two-tailed test is ±1.645.

Since the absolute value of the test statistic (0.237) is less than the critical value (1.645), we fail to reject the null hypothesis.

(d) Based on the results, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that students whose first language is not English score differently on the math portion of the SAT exam compared to students whose first language is English.

(a) It is necessary for SAT math scores to be normally distributed to test the hypotheses using the methods of this section because the methods rely on the assumption that the sample data follows a normal distribution. When the data is normally distributed, it allows for the use of parametric statistical tests, such as the t-test, which require the data to meet certain assumptions. These assumptions include normality of the data, which ensures that the test results are valid and reliable.

(b) The appropriate null and alternative hypotheses to assess whether students whose first language is not English score differently on the math portion of the SAT exam are:

Null Hypothesis (H0): There is no difference in the mean SAT math scores between students whose first language is English and students whose first language is not English.
Alternative Hypothesis (Ha): Students whose first language is not English score differently on the math portion of the SAT exam compared to students whose first language is English.

(c) To test the hypothesis, we will use the classical or P-value approach at the α = 0.1 level of significance.

First, we need to calculate the test statistic. Since the population standard deviation is known, and the sample size is small (n < 30), we will use the one-sample z-test.

The formula to calculate the test statistic (z-score) is:
z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

Using the given values:
Sample mean (x̄) = 522
Population mean (μ) = 516
Population standard deviation (σ) = 114
Sample size (n) = 20

Calculating the test statistic:
z = (522 - 516) / (114 / sqrt(20))
z = 6 / (114 / 4.472)
z = 6 / 25.338
z = 0.237

Next, we look up the critical value for a two-tailed test with α = 0.1. From the z-table, the critical value is approximately ±1.645.

Now, we compare the test statistic to the critical value. Since |0.237| < 1.645, we fail to reject the null hypothesis.

(d) Based on the results of the hypothesis test, we fail to reject the null hypothesis. This means that there is not enough evidence to conclude that students whose first language is not English score differently on the math portion of the SAT exam compared to students whose first language is English at the α = 0.1 level of significance.