Many of us have heard the stereotypical observa- tion that men absolutely refuse to stop and ask for directions, preferring instead to wander about until they “find their own way.” Is this just a stereotype, or is there really something to it?
When Lincoln Mercury was carrying out research prior to the design and introduction of its in- vehicle navigational systems, the company surveyed men and women with regard to their driving characteristics and navigational habits.
According to the survey, 61% of the female respondents said they would stop and ask for directions once they figured out they were lost.
On the other hand, only 42% of the male respon- dents said they would stop and ask for directions under the same circumstances. For purposes of our discussion, we’ll assume that Lincoln Mercury sur- veyed 200 persons of each gender.
Is it possible that the lower rate of direction- asking for men—a sample proportion of just 0.42 (84 out of 200) versus 0.61 (122 out of 200)—may have occurred simply by chance variation? Actually, if the population proportions were really the same, there would be only a 0.0001 probability of the direction-asking proportion for men being
this much lower just by chance. Section 11.6 of this chapter will provide some guidance regarding a statistical test with which you can compare these gender results and verify the conclusion for yourself. There will be no need to stop at Section 11.5 to ask for directions.
Source: Jamie LaReau, “Lincoln Uses Gender Stereotypes to Sell Navigation System,” Automotive News, May 30, 2005, p. 32.
• Select and use the appropriate hypothesis test in comparing the means of two independent samples.
• Test the difference between sample means when the samples are not independent.
• Test the difference between proportions for two independent samples.
• Determine whether two independent samples could have come from populations having the same standard deviation.
INTRODUCTION
One of the most useful applications of business statistics involves comparing two samples to examine whether a difference between them is (1) significant or (2) likely to have been due to chance. This lends itself quite well to the analysis of data from experiments such as the following:
( 11.1 )
EXAMPLE
Comparing Samples
A local YMCA, in the early stages of telephone solicitation to raise funds for expanding its gymnasium, is testing two appeals. Of 100 residents approached with appeal A, 21% pledged a donation. Of 150 presented with appeal B, 28%
pledged a donation.
SOLUTION
Is B really a superior appeal, or could its advantage have been merely the result of chance variation from one sample to the next? Using the techniques in this chap- ter, we can reach a conclusion on this and similar questions. The approach will be very similar to that in Chapter 10, where we dealt with one sample statistic (either }x or p), its standard error, and the level of significance at which it differed from its hypothesized value.
EXAMPLE EXAMPLE EXAMPLE
Sections 11.2–11.4 and 11.6 deal with the comparison of two means or two proportions from independent samples. Independent samples are those for which the selection process for one is not related to that for the other. For example, in an experiment, independent samples occur when persons are randomly assigned to the experimental and control groups. In these sections, a hypothesis-testing procedure very similar to that in Chapter 10 will be followed. As before, either one or two critical value(s) will be identified, and then a decision rule will be applied to see if the calculated value of the test statistic falls into a rejection region specified by the rule.
When comparing two independent samples, the null and alternative hypotheses can be expressed in terms of either the population parameters (1 and 2, or 1 and 2) or the sampling distribution of the difference between the sample statistics ( }x 1 and}x 2, or p1 and p2). These approaches to describing the null and alternative hypotheses are equivalent and are demonstrated in Table 11.1.
LEARNING OBJECTIVES
After reading this chapter, you should be able to:
Section 11.5 is concerned with the comparison of means for two dependent samples. Samples are dependent when the selection process for one is related to the selection process for the other. A typical example of dependent samples occurs when we have before-and-after measures of the same individuals or objects. In this case we are interested in only one variable: the difference between measure- ments for each person or object.
In this chapter, we present three different methods for comparing the means of two independent samples: the pooled-variances t-test (Section 11.2), the unequal-variances t-test (Section 11.3), and the z-test (Section 11.4). Figure 11.1 summarizes the procedure for selecting which test to use in comparing the sample means.
As shown in Figure 11.1, an important factor in choosing between the pooled-variances t-test and the unequal-variances t-test is whether we can as- sume the population standard deviations (and, hence, the variances) might be equal. Section 11.7 provides a hypothesis-testing procedure by which we can ac- tually test this possibility. However, for the time being, we will use a less rigorous standard—that is, based on the sample standard deviations, whether it appears that the population standard deviations might be equal.
THE POOLED-VARIANCES t-TEST FOR COMPARING THE MEANS OF TWO INDEPENDENT SAMPLES
Situations can arise where we’d like to examine whether the difference between the means of two independent samples is large enough to warrant rejecting the possibility that their population means are the same. In this type of setting, the alternative conclusion is that the difference between the sample means is small enough to have occurred by chance, and that the population means really could be equal.
( 11.2 )
TABLE 11.1 When comparing means from independent samples, null and alternative hypotheses can be expressed in terms of the population parameters (on the left) or described by the mean of the sampling distribu tion of the difference between the sample statistics (on the right).
This also applies to testing two sample proportions. For example, H0: 152 is the same as H0: (p12p2)5 0.
Null and Alternative Hypotheses Expressed in Hypotheses Expressed Terms of the Sampling in Terms of the Distribution of the Difference Population Means Between the Sample Means Two-tail test:
H0: 152 H0: ( }x12}x2) 5 0
or
H1: 1 2 H1: ( }x12}x2) 0 Left-tail test:
H0: 1$2 H0: ( }x12}x2) $ 0
or
H1: 1,2 H1: ( }x12}x2) , 0 Right-tail test:
H0: 1#2 H0: ( }x12}x2) # 0
or
H1: 1.2 H1: ( }x12}x2) . 0
FIGURE 11.1
Selecting the test statistic for hypothesis tests comparing the means of two independent samples.
Hypothesis test m1 – m2
Are the population standard deviations equal?
Doess1 = s2?
For any sample sizes Only if n1 and n2 both ≥ 30 No
Yes
Compute the pooled estimate of the common variance as
and perform a pooled-variances t-test where
Test assumes samples are from normal populations with equal standard deviations.
Section 11.2 See Note 1
Section 11.4 See Note 3
Az-test approximation can be performed where
withs21ands22as estimates of s21 and s22,
s2p =
s2p
(n1 – 1)s21+(n2 – 1)s22 n1 + n2– 2
t=
df = n1 + n2 – 2 and (m1 – m2)0 is from H0
and (m1 – m2)0 is from H0 The central limit theorem prevails and there are no limitations on the population distributions. This test may also be more convenient for solutions based on pocket calculators.
(x1 – x2) – (m1 – m2)0 ––1
n1 +––1 n2
√ ( )
z=(x1 – x2) – (m1 – m2)0 s21
n1 s22 n2
√ + and
Section 11.3 See Note 2 Perform an unequal-variances t-test where
[(s21 / n1) + (s22/ n2)]2
(s21 / n1)2 (s22 / n2)2 df=
and (m1 – m2)0 is from H0 Test assumes samples are from normal populations. When either test can be applied to the same data, the unequal-variances t-test is preferable to the z-test—especially when doing the test with computer assistance.
t=(x1 – x2) – (m1 – m2)0 s21
n1 s22 n2
√ +
n1 – 1 + n2 – 1
1. Section 11.7 describes a procedure for testing the null hypothesis that 152. However, when using a computer and statistical software, it may be more convenient to simply bypass this assumption and apply the unequal-variances t-test described in Section 11.3.
2. This test involves a corrected df value that is smaller than if 152 had been assumed.
When using a computer and statistical software, this test can be used routinely instead of the other two tests shown here. The nature of the df expression makes hand calculations somewhat cumbersome. The normality assumption becomes less important for larger sample sizes.
3. When sample sizes are large (each n$ 30), the z-test is a useful alternative to the unequal-variancest-test, and may be more convenient when hand calculations are involved.
4. For each test, Computer Solutions within the chapter describe the Excel and Minitab procedures for carrying out the test. Procedures based on data files and those based on summary statistics are included.
The t-test used here is known as the pooled-variances t-test because it involves the calculation of an estimated value for the variance that both populations are assumed to share. This pooled estimate is shown in Figure 11.1 as s 2 p . The number of degrees of freedom associated with the test will be df 5 n1 1 n2 2 2, and the test statistic is calculated as follows:
Our use of the t-test assumes that (1) the (unknown) population standard deviations are equal, and (2) the populations are at least approximately normally distributed. Because of the central limit theorem, the assumption of population normality becomes less important for larger sample sizes. Although it is often associated only with small-sample tests, the t distribution is appropriate when the population standard deviations are unknown, regardless of how large or small the samples happen to be.
Test statistic for comparing the means of two independent samples, 1 and 2
assumed to be equal:
t 5 ( }x 12 }x 2) 2(122)0 _____________________
ẽwwwwwws 2 p ( ___ n11 1 ___ n12 ) where
}x 1 and }x 25 means of samples 1 and 2 (122)0 5 the hypothesized difference
between the population means
n1 and n25 sizes of samples 1 and 2 s1 and s25 standard deviations of
samples 1 and 2 sp5 pooled estimate of the
common standard deviation with
s 2 p5 (n12 1)s 2 1 1(n22 1)s 2 2 _____________________
n11 n2 22 and df 5n11 n222 Confidence interval for 122:
( }x 12 }x 2) 6 ty2 ẽwwwwwws 2p ( ___ n11 1 ___ n12 )
with
5 (1 2 confidence coefficient)
The numerator of the t-statistic includes (12 2)0, the hypothesized value of the difference between the population means. The hypothesized difference is generally zero in tests like those in this chapter. The term in the denominator of the t-statistic is the estimated standard error of the difference between the sample means. It is comparable to the standard error of the sampling distribution for the sample mean discussed in Chapter 10. Also shown is the confidence interval for the difference between the population means.
EXAMPLE
Pooled-Variances t-Test
Entrepreneurs developing an accounting review program for persons preparing to take the Certified Public Accountant (CPA) examination are considering two possible formats for conducting the review sessions. A random sample of 10 stu- dents are trained using format 1, and then their number of errors is recorded for a prototype examination. Another random sample of 12 individuals are trained according to format 2, and their errors are similarly recorded for the same exami- nation. For the 10 students trained with format 1, the individual performances are 11, 8, 8, 3, 7, 5, 9, 5, 1, and 3 errors. For the 12 students trained with format 2, the individual performances are 10, 11, 9, 7, 2, 11, 12, 3, 6, 7, 8, and 12 errors.
These data are in file CX11CPA.
SOLUTION
Since the study was not conducted with directionality in mind, the appropri- ate test will be two-tail. The null hypothesis is H0: 15 2, and the alternative hypothesis is H1: 1 2. The null and alternative hypotheses may also be expressed as follows:
• Null hypothesis H0: ( }x
12}x 2) 5 0 The two review formats are equally effective.
• Alternative hypothesis H1: ( }x
12}x 2) 0 The two review formats are not equally effective.