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Chapter 10: Problem 15

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 \((m=n)\) 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
Increasing m or n decreases \(\beta\); larger n is required for smaller \(\beta\).

Step by step solution

01

Define the Problem

We need to determine how the type II error probability \(\beta\) changes in an upper-tailed hypothesis test when the sample sizes \(m\) and \(n\) increase, given that \(\mu_{1} - \mu_{2} > \Delta_{0}\).
02

Understanding Type II Error

Type II error \(\beta\) occurs when we fail to reject the null hypothesis when the alternative hypothesis is true. In a test with known \(\sigma_{1}\) and \(\sigma_{2}\), \(\beta\) is related to the power of the test.
03

Analyze Effect of Sample Size

The power of a test is given by \(1 - \beta\). As the sample sizes \(m\) or \(...\) increase, the standard error decreases, making the test statistic more sensitive to deviations from the null hypothesis, which increases the power, implying a decrease in \(\beta\).
04

Effect of Increasing m or n

Specifically, as \(m\) or \(n\) increase, the variance of the sampling distribution of the difference between means \((\mu_{1} - \mu_{2})\) decreases, leading to a smaller \(\beta\) value because it's easier to detect when \(\mu_{1} - \mu_{2} > \Delta_{0}\).
05

Equal Sample Sizes Analysis

For equal sample sizes \((m = n)\), and with a fixed \(\alpha\), increasing the sample size \(n\) increases test power, thus reducing \(\beta\).
06

Required Sample Size for Reduced 尾

To achieve a smaller \(\beta\) at a fixed level of \(\alpha\), the sample size \(n\) must be increased. The larger sample size decreases the standard error, increasing the test's sensitivity.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type II Error
Type II error, denoted as \( \beta \), occurs in hypothesis testing when the test fails to reject a false null hypothesis. This means even though the alternative hypothesis is true, the test concludes that there is not enough evidence to reject the null hypothesis. This is sometimes referred to as a "false negative."Understanding Type II error is crucial because it affects the validity of test results. Lowering \( \beta \) can reduce the likelihood of making this error, ensuring more accurate test outcomes. To decrease \( \beta \), you can:
  • Increase the sample size, which makes it easier to detect true differences.
  • Increase the significance level, although this increases the risk of a Type I error.
  • Ensure better experimental design and measurement accuracy.
In summary, managing Type II errors is essential for reliable hypothesis testing.
Power of a Test
The power of a statistical test is defined as \( 1 - \beta \). It represents the probability that the test correctly rejects a false null hypothesis. A high power indicates a high probability of detecting an effect when there is one, thus minimizing the risk of a Type II error.Power is directly related to type II error; as one decreases, the other increases. Factors that can increase the power of a test include:
  • Increasing the sample size, which reduces the standard error.
  • Selecting a higher significance level, which can increase the test's sensitivity.
  • Using a more precise measurement method to capture data accurately.
In practice, ensuring a high power level in your test design is important for achieving reliable and valid results.
Sample Size Determination
Determining the appropriate sample size is a key aspect of designing a statistical test. The sample size affects the standard error, which in turn influences the test's power and the likelihood of a Type II error.When you increase the sample size, the variability in the sampling distribution decreases. This means:
  • The test statistic becomes more reliable.
  • The standard error is reduced, increasing the test's power.
  • There's a lower probability of Type II error (\( \beta \)).
For equal sample sizes \( (m = n) \) and a fixed significance level, reducing \( \beta \) requires increasing \( n \). This ensures that the test is more sensitive to detect real differences between sample means. It's a balance of cost, time, and the desired power.
Upper-Tailed Test
An upper-tailed test is a type of hypothesis test where the critical region is in the upper tail of the distribution. It is used when the alternative hypothesis states that a parameter is greater than the null hypothesis value. In an upper-tailed test, we often focus on whether the observed data provide sufficient evidence to conclude that a sample mean is significantly greater than a hypothesized value. Important points about the upper-tailed test include:
  • It examines whether the sample mean exceeds the null hypothesis mean beyond a certain threshold.
  • The rejection region is located on the right side of the distribution.
  • It is often used in quality control and research where increases in measures are expected.
Understanding when and how to use an upper-tailed test can guide researchers in effectively testing hypotheses related to real-world phenomena.

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Most popular questions from this chapter

An experiment to compare the tension bond strength of polymer latex modified mortar (Portland cement mortar to which polymer latex emulsions have been added during mixing) to that of unmodified mortar resulted in \(\bar{x}=18.12 \mathrm{kgf} / \mathrm{cm}^{2}\) for the modified mortar \((m=40)\) and \(\bar{y}=16.87 \mathrm{kgf} / \mathrm{cm}^{2}\) for the unmodified mortar ( \(n=32\) ). Let \(\mu_{1}\) and \(\mu_{2}\) be the true average tension bond strengths for the modified and unmodified mortars, respectively. Assume that the bond strength distributions are both normal. a. Assuming that \(\sigma_{1}=1.6\) and \(\sigma_{2}=1.4\), test \(H_{0}\) : \(\mu_{1}-\mu_{2}=0\) versus \(H_{\mathrm{a}}: \mu_{1}-\mu_{2}>0\) at level \(.01\). b. Compute the probability of a type II error for the test of part (a) when \(\mu_{1}-\mu_{2}=1\). c. Suppose the investigator decided to use a level \(.05\) test and wished \(\beta=.10\) when \(\mu_{1}-\mu_{2}=1\). If \(m=40\), what value of \(n\) is necessary? d. How would the analysis and conclusion of part (a) change if \(\sigma_{1}\) and \(\sigma_{2}\) were unknown but \(s_{1}=1.6\) and \(s_{2}=1.4\) ?

Statin drugs are used to decrease cholesterol levels, and therefore hopefully to decrease the chances of a heart attack. In a British study ("MRC/BHF Heart Protection Study of Cholesterol Lowering with Simvastin in 20,536 High-Risk Individuals: A Randomized Placebo-Controlled Trial,"Lancet, 2002: 7-22) 20,536 at-risk adults were assigned randomly to take either a 40 -mg statin pill or placebo. The subjects had coronary disease, artery blockage, or diabetes. After 5 years there were 1328 deaths (587 from heart attack) among the 10,269 in the statin group and 1507 deaths (707 from heart attack) among the 10,267 in the placebo group. a. Give a \(95 \%\) confidence interval for the difference in population death proportions. b. Give a \(95 \%\) confidence interval for the difference in population heart attack death proportions. c. Is it reasonable to say that most of the difference in death proportions is due to heart attacks, as would be expected?

An experiment to determine the effects of temperature on the survival of insect eggs was described in the article "Development Rates and a TemperatureDependent Model of Pales Weevil" (Emiron. Entomol, 1987: 956-962). At \(11^{\circ} \mathrm{C}, 73\) of \(91 \mathrm{eggs}\) survived to the next stage of development. At \(30^{\circ} \mathrm{C}\), 102 of 110 eggs survived. Do the results of this experiment suggest that the survival rate (proportion surviving) differs for the two temperatures? Calculate the \(P\)-value and use it to test the appropriate hypotheses.

A study seeks to compare hospitals based on the performance of their intensive care units. The dependent variable is the mortality ratio, the ratio of the number of deaths over the predicted number of deaths based on the condition of the patients. The comparison will be between hospitals with nurse staffing problems and hospitals without such problems. Assume, based on past experience, that the standard deviation of the mortality ratio will be around \(.2\) in both types of hospital. How many of each type of hospital should be included in the study in order to have both the type I and type II error probabilities be \(.05\), if the true difference of mean mortality ratio for the two types of hospital is .2? If we conclude that hospitals with nurse staffing problems have a higher mortality ratio, does this imply a causal relationship? Explain.

Is the response rate for questionnaires affected by including some sort of incentive to respond along with the questionnaire? In one experiment, 110 questionnaires with no incentive resulted in 75 being retumed, whereas 98 questionnaires that included a chance to win a lottery yielded 66 responses ("Charities, No; Lotteries, No; Cash, Yes," Public Opinion Q., 1996: 542-562). Does this data suggest that including an incentive increases the likelihood of a response? State and test the relevant hypotheses at significance level \(.10\) by using the \(P\)-value method.
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