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Chapter 9: Q15 E (page 372)

a. Show for the upper-tailed test with \({\sigma _1}\) and \({\sigma _2}\)known that as either\(m\) or\(n\) increases, \(\beta \)decreases when \({\mu _1} - {\mu _2} > {\Delta _0}\).

b. For the case of equal sample sizes \(\left( {m = n} \right)\)and fixed \(\alpha \),what happens to the necessary sample size \(n\) as \(\beta \) is decreased, where \(\beta \) is the desired type II error probability at a fixed alternative?

Short Answer

Expert verified

the solution is

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).

Step by step solution

01

show the upper-tailed test

\({H_0}:{\mu _1} - {\mu _2} = {\Delta _0}\)represents the null hypothesis. The type II error \(\beta \) for \({\mu _1} - {\mu _2} = {\Delta ^'}\) vary depending on the alternative hypothesis. The alternative hypothesis is \({H_a}:{\mu _1} - {\mu _2} > 0\), which indicates that the type II error is present

\(\beta \left( {{\Delta ^'}} \right) = \Phi \left( {{z_\alpha } - \frac{{\Delta ' - {\Delta _0}}}{\sigma }} \right)\)

Where

\(\sigma = \sqrt {\frac{{\sigma _1^2}}{m} + \frac{{\sigma _2^2}}{n}} \)

When \(n\) or \(m\) goes up, \(\sigma \) goes down (because we divide with bigger number). This implies that.

\(\frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }\)

Because the numerator is positive, the value will rise. As a result, the number.

\({z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }\)

Because \(\Phi \) is a cdf of standard normal distribution, the type Il error lowers.

\(\Phi \left( {{z_\alpha } - \frac{{{\Delta ^'} - {\Delta _0}}}{\sigma }} \right)\)

Will decreases as well.

02

type II error probability

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).

03

conclusion

The corresponding \({z_\beta }\) increases as the intended \(\beta \) (type II error) lowers. When the sample sizes are equivalent, \({z_\beta }\) is in the numerator of the formula for necessary sample size \(n\), indicating that when \({z_\beta }\) rises, the sample size increases as well (see exercise 13).

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Most popular questions from this chapter

The article "Fatigue Testing of Condoms" cited in Exercise 7.32 reported that for a sample of 20 natural latex condoms of a certain type, the sample mean and sample standard deviation of the number of cycles to break were 4358 and 2218 , respectively, whereas a sample of 20 polyisoprene condoms gave a sample mean and sample standard deviation of 5805 and 3990 , respectively. Is there strong evidence for concluding that true average number of cycles to break for the polyisoprene condom exceeds that for the natural latex condom by more than 1000 cycles? Carry out a test using a significance level of 01 . (Note: The cited paper reported P-values of t tests for comparing means of the various types considered.)

Sometimes experiments involving success or failure responses are run in a paired or before/after manner. Suppose that before a major policy speech by a political candidate, n individuals are selected and asked whether \((S)\)or not (F) they favor the candidate. Then after the speech the same n people are asked the same question. The responses can be entered in a table as follows:

Before

After

S

F

S

\({{\bf{X}}_{\bf{1}}}\)

\({{\bf{X}}_{\bf{2}}}\)

F

\({{\bf{X}}_{\bf{3}}}\)

\({{\bf{X}}_{\bf{4}}}\)

Where\({{\bf{x}}_{\bf{1}}}{\bf{ + }}{{\bf{x}}_{\bf{2}}}{\bf{ + }}{{\bf{x}}_{\bf{3}}}{\bf{ + }}{{\bf{x}}_{\bf{4}}}{\bf{ = n}}\). Let\({{\bf{p}}_{\bf{1}}}{\bf{,}}{{\bf{p}}_{\bf{2}}}{\bf{,}}{{\bf{p}}_{\bf{3}}}\), and \({p_4}\)denote the four cell probabilities, so that \({p_1} = P(S\) before and S after), and so on. We wish to test the hypothesis that the true proportion of supporters (S) after the speech has not increased against the alternative that it has increased.

a. State the two hypotheses of interest in terms of\({p_1},{p_2}\),\({p_3}\), and \({p_4}\).

b. Construct an estimator for the after/before difference in success probabilities

c. When n is large, it can be shown that the rv \(\left( {{{\bf{X}}_{\bf{i}}}{\bf{ - }}{{\bf{X}}_{\bf{j}}}} \right){\bf{/n}}\) has approximately a normal distribution with variance given by\(\left[ {{{\bf{p}}_{\bf{i}}}{\bf{ + }}{{\bf{p}}_{\bf{j}}}{\bf{ - }}{{\left( {{{\bf{p}}_{\bf{i}}}{\bf{ - }}{{\bf{p}}_{\bf{j}}}} \right)}^{\bf{2}}}} \right]{\bf{/n}}\). Use this to construct a test statistic with approximately a standard normal distribution when \({H_0}\)is true (the result is called McNemar's test).

d. If\({{\bf{x}}_{\bf{1}}}{\bf{ = 350,}}\;\;\;{{\bf{x}}_{\bf{2}}}{\bf{ = 150,}}\;\;\;{{\bf{x}}_{\bf{3}}}{\bf{ = 200}}\), and\({x_4} = 300\), what do you conclude?

Consider the pooled\(t\)variable

\(T = \frac{{(\bar X - \bar Y) - \left( {{\mu _1} - {\mu _2}} \right)}}{{{S_p}\sqrt {\frac{1}{m} + \frac{1}{n}} }}\)

which has a\(t\)distribution with\(m + n - 2\)df when both population distributions are normal with\({\sigma _1} = {\sigma _2}\)(see the Pooled\(t\)Procedures subsection for a description of\({S_p}\)).

a. Use this\(t\)variable to obtain a pooled\(t\)confidence interval formula for\({\mu _1} - {\mu _2}\).

b. A sample of ultrasonic humidifiers of one particular brand was selected for which the observations on maximum output of moisture (oz) in a controlled chamber were\(14.0, 14.3, 12.2\), and 15.1. A sample of the second brand gave output values\(12.1, 13.6\),\(11.9\), and\(11.2\)("Multiple Comparisons of Means Using Simultaneous Confidence Intervals," J. of Quality Technology, \(1989: 232 - 241\)). Use the pooled\(t\)formula from part (a) to estimate the difference between true average outputs for the two brands with a\(95\% \)confidence interval.

c. Estimate the difference between the two\(\mu \)'s using the two-sample\(t\)interval discussed in this section, and compare it to the interval of part (b).

Refer to Example 9.7. Does the data suggest that the standard deviation of the strength distribution for fused specimens is smaller than that for not-fused specimens? Carry out a test at significance level .01.

How does energy intake compare to energy expenditure? One aspect of this issue was considered in the article 鈥淢easurement of Total Energy Expenditure by the Doubly Labelled Water Method in Professional Soccer Players鈥� (J. of Sports Sciences, 2002: 391鈥�397), which contained the accompanying data (MJ/day).

Test to see whether there is a significant difference between intake and expenditure. Does the conclusion depend on whether a significance level of .05, .01, or .001 is used?
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