Question

Test Statistics Identify the test statistic that should be used for testing the following given claims.

a. The mean of the differences between IQ scores of husbands and IQ scores of their wives is

equal to 0.

b. The mean age of female CIA agents is equal to the mean age of male CIA agents.

c. The proportion of left-handed men is equal to the proportion of left-handed women.

d. The variation among pulse rates of women is equal to the variation among pulse rates of men.

Short answer

The test statistics are given below:

a.\(t = \frac{{\bar d - {\mu _d}}}{{\frac{{{s_d}}}{{\sqrt n }}}}\)

b.\(t = \frac{{\left( {{{\bar x}_F} - {{\bar x}_M}} \right) - \left( {{\mu _F} - {\mu _M}} \right)}}{{\sqrt {\frac{{s_F^2}}{{{n_F}}} + \frac{{s_M^2}}{{{n_M}}}} }}\)

c.\(t = \frac{{\left( {{{\hat p}_M} - {{\hat p}_W}} \right) - \left( {{p_M} - {p_W}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_M}}} + \frac{{\bar p\bar q}}{{{n_W}}}} }}\)

d. \(F = \frac{{s_W^2}}{{s_M^2}}\)

Step-by-step solution

Given information

Four cases are provided for the hypotheses test.

State the method to identify test statistic

The following steps are followed for determining the test statistic for any hypotheses test.

1. Determine the parameter(s) being tested or compared.
2. Note the number of groups being compared and check if they are independent or not.
3. Check the underlying conditions of the study like normality, equal variance, etc.
4. Note the measures known in the study.
5. The test statistic is determined based on this knowledge.

Identify test statistics in each case

a.

Claim: Test that the mean difference of IQ scores between husbands and wives is 0.

The parameters being comparedare themeans of two populations of husbands and wives. The IQ scores in a population can be assumed to be normally distributed.

Note that the groups have paired observations as husbands and wives are related. Thus, it can be inferred that the two-sample paired t-test is used for testing the claim.

The related test statistic is given below:

\(t = \frac{{\bar d - {\mu _d}}}{{\frac{{{s_d}}}{{\sqrt n }}}}\)

Here,\(\bar d,{\mu _d},{s_d},n\)are sample mean difference, true mean difference, standard deviation of sample differences, and the count of paired observations, respectively.

b.

Claim: Compare the mean age of female and male CIA agents

The parameters being comparedare themeans of two groups of female and male CIA agents. The population of ages can be assumed to be normally distributed.

Note that the groups are independent. Assume that only sample statistics are known, and sample standard deviations of groups are unequal. Thus, it can be inferred that the two-sample independent t-test willbe used for testing the claim.

The related test statistic is given below:

\(t = \frac{{\left( {{{\bar x}_F} - {{\bar x}_M}} \right) - \left( {{\mu _F} - {\mu _M}} \right)}}{{\sqrt {\frac{{s_F^2}}{{{n_F}}} + \frac{{s_M^2}}{{{n_M}}}} }}\)

Here,\({\bar x_F},{\mu _F},s_F^2,{n_F}\)are sample mean, true population mean, sample standard deviation, count of observations, respectively, for female CIA agents.

And\({\bar x_M},{\mu _M},s_M^2,{n_M}\)are sample mean, true population mean, sample standard deviation, count of observations, respectively, for male CIA agents.

c.

Claim: Compare the proportion of left-handed men and women

The parameters being compared are theproportion of two groups ofmen and women who are left-handed. The populationsare normally distributed.

Note that the groups are compared for equality of proportions. Thus, it can be inferred that the two sample z-test for proportionswillbe used for testing the claim.

The related test statistic is given below:

\(t = \frac{{\left( {{{\hat p}_M} - {{\hat p}_W}} \right) - \left( {{p_M} - {p_W}} \right)}}{{\sqrt {\frac{{\bar p\bar q}}{{{n_M}}} + \frac{{\bar p\bar q}}{{{n_W}}}} }}\)

Here,\({\hat p_W},{p_W},{n_W}\)are sample proportions, true population proportions, count of subjects in groups, respectively, for left-handed women.

And\({\hat p_M},{p_M},{n_M}\)are sample proportions, true population proportions, count of subjects in groups, respectively, for left-handed men.

Also,\(\bar p\)is the pooled proportion.

d.

Claim: Compare the variances between pulse rates of women and men

The parameters being comparedare thevariances of two groups ofpulse rates men and women. The populations of pulse rates are normally distributed.

Note that the groups are compared for equality of variances. Thus, it can be inferred that the F-test for variances willbe used for testing the claim.

The related test statistic is given below:

\(F = \frac{{s_W^2}}{{s_M^2}}\)

Here, \(s_W^2,s_M^2\)are sample variances of the pulse rates for women and men.