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Chapter 9: Problem 59

Two sampling distributions A study is designed to test \(\mathrm{H}_{0}: p=0.50\) against \(\mathrm{H}_{a}: p>0.50,\) taking a random sample of size \(n=100,\) using significance level \(0.05 .\) a. Show that the rejection region consists of values of \(\hat{p}>0.582\) b. Sketch a single picture that shows (i) the sampling distribution of \(\hat{p}\) when \(\mathrm{H}_{0}\) is true and (ii) the sampling distribution of \(\hat{p}\) when \(p=0.60 .\) Label each sampling distribution with its mean and standard error, and highlight the rejection region. c. Find \(\mathrm{P}\) (Type II error) when \(p=0.60\).

Short Answer

Expert verified
a. Rejection region: \(\hat{p} > 0.582\). b. Sketch includes distributions for \(\mathrm{H}_0\) (mean=0.50) and \(p=0.60\) (mean=0.60), showing rejection region. c. Type II error probability (\(\beta\)) is approximately 0.3577.

Step by step solution

01

Determine the Critical Value

First, calculate the standard error (SE) of the sampling distribution of \(\hat{p}\) under \(\mathrm{H}_0\): \(p = 0.50\). \[SE = \sqrt{\frac{p(1-p)}{n}} = \sqrt{\frac{0.50 \times 0.50}{100}} = 0.05\]Next, find the critical value for \(\hat{p}\) such that the probability of being above this value is 0.05. This corresponds to finding \(z\)-value:\[ \text{Using } z_{0.05} = 1.645 \]Then, calculate the critical value:\[\hat{p}_c = p + z \cdot SE = 0.50 + (1.645 \times 0.05) = 0.582\]Thus, the rejection region consists of values \(\hat{p} > 0.582\).
02

Sketch Sampling Distributions

Draw two normal distributions on the same axes:1. **Distribution under \(\mathrm{H}_0\)**: - Mean = 0.50 - Standard Error (SE) = 0.05 - Rejection region \(\hat{p} > 0.582\) shaded to the right of 0.582.2. **Distribution when \(p = 0.60\)**: - Mean = 0.60 - Standard Error (SE) = 0.049 (alternative SE calculated for \(p = 0.60\)) - Overlay this distribution on the same graph.Label each distribution clearly with mean, SE, and highlight the rejection region for \(\mathrm{H}_0\).
03

Calculate Probability of Type II Error

Type II error (P(Beta)) occurs when \(\mathrm{H}_0\) is not rejected given \(p = 0.60\).Calculate \(z\)-score for \(\hat{p} = 0.582\) using the sampling distribution for \(p = 0.60\):\[z = \frac{0.582 - 0.60}{0.049} = -0.367\]Find the probability \(P(\hat{p} \leq 0.582 \mid p = 0.60)\) by looking at the standard normal distribution table for \(z = -0.367\), which gives a probability of approximately 0.3577. Thus, \(\beta = 0.3577\).

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Key Concepts

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

Hypothesis Testing
Hypothesis Testing is a fundamental statistical method used to determine the validity of a claim about a population parameter based on sample data. It involves setting up two competing hypotheses. In our exercise, the null hypothesis (\(H_0\)) states that the true population proportion \(p = 0.50\). The alternative hypothesis (\(H_a\)) proposes that the true population proportion is greater than 0.50 (\(p > 0.50\)).
The goal of hypothesis testing is to make a decision regarding these hypotheses based on the sample data. Generally, you reject \(H_0\) if there is sufficient evidence in the sample to support \(H_a\). In our exercise, the critical value was determined, providing a rejection region. If the sample proportion \(\hat{p}\) is greater than this critical value, \(H_0\) is rejected in favor of \(H_a\).
  • Setup both \(H_0\) and \(H_a\).
  • Choose a significance level (typically 0.05).
  • Calculate the test statistic and compare it to critical values or use p-values.
  • Make a decision to reject or not reject \(H_0\).
Significance Level
The Significance Level, denoted as \(\alpha\), is the probability of rejecting the null hypothesis when it is actually true. It reflects how much risk of making a Type I error (false positive) you are willing to accept. In most statistical tests, a significance level of 0.05 is commonly used, implying a 5% risk of incorrectly rejecting \(H_0\).
In our exercise, the significance level is set at 0.05. This level helps establish the criteria for the rejection region, defining the critical value above which we have sufficient evidence to reject \(H_0\).
  • The lower the significance level, the stronger the evidence must be to reject \(H_0\).
  • A significance level of 0.01 is stricter compared to 0.05.
  • Choosing \(\alpha\) involves consideration of the context and consequences of Type I errors.
Type II Error
A Type II Error occurs when the null hypothesis is not rejected even though it is false. In our scenario, this means failing to reject \(H_0: p = 0.50\) when in fact \(p = 0.60\). The probability of making a Type II Error is represented by \(\beta\).
In the exercise, we calculate \(\beta\) using the sampling distribution when \(p = 0.60\). A Type II error probability can be influenced by sample size, significance level, and the true parameter being tested.
  • \(\beta\) is generally reduced by increasing the sample size.
  • Maintaining a balance between \(\alpha\) and \(\beta\) is critical.
  • A small \(\beta\) means a higher likelihood of correctly rejecting a false \(H_0\).
Standard Error
Standard Error (SE) measures the variability of a sampling distribution. It is crucial for inference as it reflects how much the sample statistic (such as \(\hat{p}\)) varies from the population parameter (\(p\)). In our example, the SE is calculated to assess variability under the null hypothesis, with the formula \(SE = \sqrt{\frac{p(1-p)}{n}}\).
The SE helps determine the critical values and rejection regions in hypothesis testing. In our exercise, it was used to establish the normal distribution of the sample proportion \(\hat{p}\).
  • SE decreases with increasing sample size.
  • Smaller SE indicates more precise estimates of the population parameter.
  • SE is instrumental in calculating confidence intervals and conducting hypothesis tests.

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Most popular questions from this chapter

Gender bias in selecting managers Exercise 9.19 tested the claim that female employees were passed over for management training in favor of their male colleagues. Statewide, the large pool of more than 1000 eligible employees who can be tapped for management training is \(40 \%\) female and \(60 \%\) male. Let \(p\) be the probability of selecting a female for any given selection. For testing \(\mathrm{H}_{0}: p=0.40\) against \(\mathrm{H}_{a}: p<0.40\) based on a random sample of 50 selections, using the 0.05 significance level, verify that: a. A Type II error occurs if the sample proportion falls less than 1.645 standard errors below the null hypothesis value, which means that \(\hat{p}>0.286\). b. When \(p=0.20,\) a Type II error has probability 0.06 .

P-value Indicate whether each of the following P-values gives strong evidence or not especially strong evidence against the null hypothesis. a. 0.38 b. 0.001

Dogs and cancer A recent study \(^{6}\) considered whether dogs could be trained to detect if a person has lung cancer or breast cancer by smelling the subject's breath. The researchers trained five ordinary household dogs to distinguish, by scent alone, exhaled breath samples of 55 lung and 31 breast cancer patients from those of 83 healthy controls. A dog gave a correct indication of a cancer sample by sitting in front of that sample when it was randomly placed among four control samples. Once trained, the dogs' ability to distinguish cancer patients from controls was tested using breath samples from subjects not previously encountered by the dogs. (The researchers blinded both dog handlers and experimental observers to the identity of breath samples.) Let \(p\) denote the probability a dog correctly detects a cancer sample placed among five samples, when the other four are controls. a. Set up the null hypothesis that the dog's predictions correspond to random guessing. b. Set up the alternative hypothesis to test whether the probability of a correct selection differs from random guessing. c. Set up the alternative hypothesis to test whether the probability of a correct selection is greater than with random guessing. d. In one test with 83 Stage I lung cancer samples, the dogs correctly identified the cancer sample 81 times. The test statistic for the alternative hypothesis in part \(c\) was \(z=17.7\). Report the P-value to three decimal places, and interpret. (The success of dogs in this study made researchers wonder whether dogs can detect cancer at an earlier stage than conventional methods such as MRI scans.)

A person who claims to be psychic says that the probability \(p\) that he can correctly predict the outcome of the roll of a die in another room is greater than \(1 / 6,\) the value that applies with random guessing. If we want to test this claim, we could use the data from an experiment in which he predicts the outcomes for \(n\) rolls of the die. State hypotheses for a significance test, letting the alternative hypothesis reflect the psychic's claim.

\(\mathbf{H}_{0}\) or \(\mathbf{H}_{a}\) ? For each of the following, is the statement a null hypothesis or an alternative hypothesis? Why? a. The mean IQ of all students at Lake Wobegon High School is larger than 100 . b. The probability of rolling a 6 with a particular die equals \(1 / 6\). c. The proportion of all new business enterprises that remain in business for at least five years is less than 0.50 .
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