Suppose that in a particular geographic region, the mean and standard deviation of scores on a reading test are 100 points, and 12 points, respectively. Our interest is in the scores of 55 students in a particular school who received a mean score of 96. We can ask whether this mean score is significantly lower than the regional mean - that is, are the students in this school comparable to a simple random sample of 55 students from the region as a whole or are their scores surprisingly low?

a) Which test should used and why?

b) Perform an appropriate test and give your conclusion.

 

a) In this case the population standard deviation is known. As we are testing for single sample mean, so assuming that the population follows a normal distribution, we can use the Z test for single sample mean. For a normal distribution where the population variance (or standard deviation is known) we use the Z test for mean because the sampling distribution of the sample mean is normal. Thus Z test is appropriate in this case.

b) Here the interest is to test whether the mean score is significantly lower than the regional mean of 100 or not. So the hypotheses are,

H₀: µ ≥ 100 and H₁: µ < 100

We can see, this is a left tailed test for mean. As this is a Z test so assuming the usual 0.05 significance level the rejection region is,

Test statistic < Critical value = - Z_0.05 = -1.645

Now,

Test statistic = (x?-100)/(σ/√n) = (96-100)/(12/√55) = -2.472

As the test statistic is falling within the rejection region so the null hypothesis is rejected. The conclusion is that, the data is providing enough evidence at 0.05 significance level to conclude that the mean score of this particular geographic region is significantly lower than the regional mean of 100.