Suppose we want to assess the effect of a one-day SAT prep class at a 5% level of significance. Scores on the SAT writing exam can range from 200 to 800. A random sample of 50 students takes the SAT writing test before and after a prep class.

We test the hypotheses:
- [tex] H_0: \mu = 0 [/tex]
- [tex] H_a: \mu > 0 [/tex]

where [tex] \mu [/tex] is the mean of the difference in SAT writing scores (after minus before) for all students who take the SAT prep class. The sample mean is 5 with a standard deviation of 18.

Since the sample size is large, we are able to conduct the T-Test. The T-test statistic is approximately 1.96 with a P-value of approximately 0.028.

What can we conclude?

The SAT prep class has no influence on the mean difference in SAT writing scores, hence the null-hypothesis (H0) states that the mean difference is zero.

The alternative theory (Ha) states that the SAT prep course has a positive impact on the mean difference in SAT writing scores, resulting in a mean difference that is greater than zero.

The sample size of 50 is sufficient for us to do the hypothesis test using the t-distribution.

The estimated t-test statistic is 1.96, and at the 5% level of significance, it is significant only if it is in the rejection zone of the null hypothesis (1.677 is the crucial value for a one-tailed test with 49 degrees of freedom).

The calculated p-value of 0.028 is less than the threshold of 0.05, the null hypothesis is also rejected in favour of the alternative hypothesis.

To draw the conclusion that the SAT prep course has a favourable impact on the mean difference in SAT writing scores. Particularly, at the 5% level of significance, the sample-mean difference of 5 is statistically significantly greater than zero.

Therefore, it is reasonable.

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