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

σ= 100. What is the value of your test statistic?

(c) Let α= 0.05. What is the p-value? What is your conclusion?

(d) Let α= 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) Null Hypothesis (H₀):

"There is no significant difference between the the historical mean and the new mean". So, new mean, = 900

This is the hypothesis that the assistant Dean is testing each year.

(b) Test statistic(Z):

Given: sample mean, = 875

Population std. deviation, = 100

Sample size, n = 75 (>30: large sample)

Z = ( - )/(/) = (875 - 900)/(100/) = - 2.165

Thus, test statistic, Z= -2.165

(c)

= significance level = 0.05

p-value (two-tailed test) at = 0.05 for Z = - 2.165 is P = 0.03

Conclusion: P < i.e., 0.03 < 0.05. Thus, there is a sufficient evidence to reject the null hypothesis (H₀). Thus, there exists a statistically significant difference between the historical mean and the new mean. So, 900

(d)

Critical value of Z at = 0.05 (two-tailed test) is Z_crit = 1.96

Calculated Z- value is 2.165 (positive value must be considered here).

Since, calculated Z - value (test statistic) is more than the critical value, we reject null hypothesis. 2.165>1.96. Thus, there exists a statistically significant difference between the the historical mean and the new mean. So, 900

(e)

Since, it is proved that the new mean is not 900, we can say the that the mean application score for freshman has changed this year.