An English teacher has been teaching a sixth grade composition class for many years. He has the feeling that over the past several years, the writing ability of students has changed. A national test of proficiency in composition was administered 5 years ago. The resulting distribution of scores was normally shaped, had a mean of 85 and a standard deviation of 10.9. In order to test his feeling, he gives his present class of 43 students the same proficiency test. The resulting mean is 80 and the standard deviation is 8.7. The obtained value of the appropriate statistic for testing H 0 is _________.
A) z obt = - 0.46
B) z obt = +3.01
C) z obt = - 3.01
D) z obt = - 0.45

Question

Quizplus · Accepted Answer

We are testing if there has been a significant change in the writing ability of students over the past 5 years, so our null hypothesis is:
H0: μ = 85 (there has been no change)

Our alternative hypothesis is:
Ha: μ < 85 (there has been a decrease in writing ability)

We are given the sample statistics:
n = 43
x̄ = 80
s = 8.7

To test our hypothesis, we can use a one-sample t-test, since the population standard deviation is unknown. The test statistic is given by:

t = (x̄ - μ) / (s / sqrt(n))

Plugging in the values, we get:
t = (80 - 85) / (8.7 / sqrt(43)) = -3.01

Looking up the t-distribution table with 42 degrees of freedom and a significance level of 0.05 (since it's a two-tailed test), we find the critical value to be -1.682. Since our test statistic (-3.01) is less than the critical value (-1.682), we reject the null hypothesis and conclude that there has been a significant decrease in writing ability over the past 5 years.

However, since the options in the question are all z-scores, we need to convert our t-score to a z-score using the formula:
z = (t - μt) / σt
where μt and σt are the mean and standard deviation of the t-distribution, respectively. Since we have 42 degrees of freedom, we can use the approximation that the t-distribution is approximately normal, so μt = 0 and σt ≈ 1.

Plugging in the values, we get:
z = (-3.01 - 0) / 1 ≈ -3.01

Therefore, the obtained value of the appropriate z-statistic for testing H0 is z obt = -3.01, which corresponds to option C.

An English Teacher Has Been Teaching a Sixth Grade Composition