Hypothesis tests about the difference between two population means

defective in the sample is not significant. We have no evidence to reject the null hypothesis that the proportion defective is .01 at the 5% level of significance. The probability of our having made a Type II error (accepting H0 when, in fact, it is not true) is β = .05.

[Note that the interval

pˆ±2 pˆqˆ/n=.133±2 (.133)(1−.133)/150=.133±.056

does not contain 0 or 1. Thus, the sample size is large enough to guarantee that validity of the hypothesis test.]

Although small-sample procedures are available for testing hypotheses about a population proportion, the details are omitted from our discussion. It is our experience that they are of limited utility since most surveys of binomial population performed in the reality use samples that are large enough to employ the techniques of this section.

9.4 Hypothesis tests about the difference between two population means

There are two brands of coffee, A and B. Suppose a consumer group wishes to determine whether the mean price per pound of brand A exceeds the mean price per pound of brand B.

That is, the consumer group will test the null hypothesis

H0: (µ1 - µ2) = 0 against the alternative ((µ1 - µ2) > 0. The large-sample procedure described in the box is applicable testing a hypothesis about (µ1 - µ2), the difference between two population means.

2. The samples are selected randomly and independent from the target populations.

Example 9.6 A consumer group selected independent random samples of supper-markets located throughout a country for the purpose of comparing the retail prices per pound of coffee of brands A and B. The results of the investigation are summarized in Table 9.1. Does this evidence indicate that the mean retail price per pound of brand A coffee is significantly higher than the mean retail price per pound of brand B coffee? Use a significance level of αααα ^{= .01.}

Table 9.1 Coffee prices for Example 9.6

Brand A Brand B

Solution The consumer group wants to test the hypotheses H0: (µ1 - µ2) = 0 (i.e., no difference between mean retail prices)

Ha: (µ1 - µ2) > 0 (i.e., mean retail price per pound of brand A is higher than that of brand

We compute the test statistic as follows:

2.947

Since this computed value of z = 2.947 lies in the rejection region, there is sufficient evidence (at α = .01) to conclude that the mean retail price per pound of brand A coffee is significantly higher than the mean retail price per pound of brand B coffee. The probability of our having committed a Type I error is α = .01.

When the sample sizes n1 and n2 are inadequate to permit use of the large-sample procedure of Example 9.9, we have made some modifications to perform a small-sample test of hypothesis about the difference between two population means. The test procedure is based on assumption that are more restrictive than in the large-sample case. The elements of the hypothesis test and required assumption are listed in the next box.

Small-sample test of hypothesis about (µµµµ1 - µµµµ2)

Test statistic:

1. The population from which the samples are selected both have approximately normal relative frequency distributions.

2. The variances of the two populations are equal.

3. The random samples are selected in an independent manner from the two populations.

Example 9.7 There was a research on the weights at birth of the children of urban and rural women. The researcher suspects there is a significant difference between the mean weights at birth of children of urban and rural women. To test this hypothesis, he selects independent random samples of weights at birth of children of mothers from each group, calculates the mean weights and standard deviations and summarizes in Table 9.2. Test the researcher's belief, using a significance of αααα ^{= .02.}

Table 9.2 Weight at birth data for Example 9.7 Urban mothers Rural mothers

n₁ = 15

Solution The researcher wants to test the following hypothesis:

H₀: (µ1 - µ2) = 0 (i.e., no difference between mean weights at birth)

Ha: (µ1 - µ2) ≠ 0 (i.e., mean weights at birth of children of urban and rural women are different)

where µ1 and µ2 are the true mean weights at birth of children of urban and rural women, respectively.

Since the sample sizes for the study are small (n1 = 15, n2 = 14), the following assumptions are required:

1. The populations of weights at birth of children both have approximately normal distributions.

2. The variances of the populations of weights at birth of children for two groups of mothers are equal.

3. The samples were independently and randomly selected.

If these three assumptions are valid, the test statistic will have a t-distribution with (n1 + n2 - 2) = (15 + 14 - 2) = 27 degree of freedom with a significance level of α = .02, the rejection region is given by

t < - t.01 = - 2.473 or t > t.01 = 2.473 (see Figure 9.4)

Figure 9.4 Rejection region of Example 9.7

Since we have assumed that the two populations have equal variances (i.e. that σ

σ

σ₁² = ₂² = ), we need to compute an estimate of this common variance. Our pooled estimate is given by

Using this pooled sample variance in the computation of the test statistic, we obtain

2.422

Now the computed value of t does not fall within the rejection region; thus, we fail to reject the null hypothesis (at α = .02) and conclude that there is insufficient evidence of a difference between the mean weights at birth of children of urban and rural women.

In this example, we can see that the computed value of t is very closed to the upper boundary of the rejection region. This region is specified by the significance level and the degree of freedom.

How is the conclusion about the difference between the mean weights at births affected if the significance level is α = .05? We will answer the question in the next example.

Example 9.8 Refer Example 9.7. Test the investigator's belief, using a significance level of αααα = .05.

Solution With a significance level of αααα = .05, the rejection region is given by t < - t.025 = - 2.052 or t > t.025 = 2.052 (see Figure 9.5)

Since the sample sizes are not changed, therefore test statistic is the same as in Example 9.10, t = 2.422.

Now the value of t falls in the rejection region; and we have sufficient evidence at a significance level of α = .05 to conclude that the mean weight at birth of children of urban women differs significantly (or we can say that is higher than) from the mean weight at birth of children of rural women. But you should notice that the probability of our having committed a Type I error is α ⁼ .05.

Figure 9.5 Rejection region of Example 9.8