Question

- At Western University the historical mean of scholarship examination scores for freshman applications is 900. Each year the assistant dean uses a sample of applications to determine whether the mean examination score for new freshman applications has changed. (a) What hypotheses is the assistant dean testing each year? (b) This year for a sample of 75 freshman, the dean finds the sample average examination score to be 875. The dean also knows that 0 = 100. What is the value of your test statistic? (c) Let a = 0.05. What is the p-value? What is your conclusion? (d) Let a = 0.05. What is the critical value? What is your conclusion (this should be the same as the previous part)? (e) Has the mean application score for freshman changed this year?

Answer 1

(a)
H0: Null Hypothesis: $\mu$ = 900 (The mean examination score for new freshman applications has not changed)

HA: Alternative Hypothesis: $\mu$ $\neq$ 900 (The mean examination score for new freshman applications has changed) (Claim)

(b)

n = 75

$\bar{}$$\bar{x}$ = 875

$\sigma$ = 100

SE = $\sigma$ /$\sqrt{n}$

= 100/$\sqrt{75}$

= 11.5470

Test Statistic is given by:

Z = (875 - 900)/11.5470

= - 2.1651

(c)

Table of Area Under Standard Normal Curve gives area = 0.4850

So,

p - value = (0.5 - 0.4850) X 2 = 0.03

Since p - value = 0.03 is less than $\alpha$ = 0.05, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the mean examination score for new freshman applications has changed.

(d)

$\alpha$ = 0.05

From Table, critical values of Z = $\pm$ 1.96

Since calculated value of Z = - 2.1651 is less than critical value of Z = - 1.96, the difference is significant. Reject null hypothesis.

Conclusion:
The data support the claim that the mean examination score for new freshman applications has changed.

(e)

The mean examination score for new freshman applications has changed.