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Chapter 9: Problem 72

Consider the following null and alternative hypotheses: $$ H_{0}=p=.44 \text { versus } H_{1}: p<.44 $$ A random sample of 450 observations taken from this population produced a sample proportion of \(.39 .\) a. If this test is made at a \(2 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01 ?\) What if \(\alpha=.025\) ?

Short Answer

Expert verified
Without actual calculation, we cannot provide definitive outcomes. Here's an overview: (a) If the calculated z-score in step 2 is less than -2.05, then we reject the null hypothesis. Otherwise, we fail to reject it. (b) The probability of making a Type I error is 2%. (c) If the p-value calculated in step 5 is less than .01, we reject the null hypothesis at \(\alpha = .01\), and if it's less than .025, we reject the null hypothesis at \(\alpha = .025\).

Step by step solution

01

Calculating a Z-score

Start by calculating the z-score using the formula: \(z = \frac{p - P_o}{\sqrt{\frac{P_o(1 - P_o)}{n}}}\), where \(n\) is the sample size, \(P_0\) represents the hypothesized proportion in the null hypothesis and \(p\) is the sample proportion. In our case, \(P_o = 0.44\), \(n = 450\) and \(p = 0.39\).
02

Obtain critical z-value

Next, obtain the critical z-value for the given significance level (2%). As our alternate hypothesis is \(p < .44\), this is a one-tailed test. The critical z-value for a one-tailed test at a 2% significance level is -2.05.
03

Decision

Compare the calculated z-value with the critical z-value. If the calculated z-value is less than the critical z-value, reject the null hypothesis.
04

Probability of Type I error

The probability of making a Type I error (rejecting true null hypothesis) is the significance level of the test, which is 0.02 or 2% in this case.
05

Calculate P-value

This is the probability that the z-score is less than the calculated z-value. Calculate this using a standard normal cumulative distribution table or relevant function in statistical software.
06

Decision based on P-value

If the P-value is less than the significance level \(\alpha\), then reject the null hypothesis. We compare the calculated P-value with \(\alpha = .01\) and \(\alpha = .025\).

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Key Concepts

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

Null Hypothesis
The null hypothesis, often denoted as \( H_0 \), is a foundational concept in hypothesis testing. It represents the default or initial assumption about a population parameter. In this context, the null hypothesis is that the population proportion \( p \) is equal to 0.44.
This means we assume there is no change or effect unless there is strong evidence against it.
When conducting a hypothesis test, the null hypothesis is what you generally seek evidence to reject. It serves as a baseline that will be tested with sample data through statistical methods.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \), is the statement you aim to support through your hypothesis test. It represents a change or effect contrary to the null hypothesis.
In this example, the alternative hypothesis is \( p < 0.44 \). This suggests that the sample proportion is indeed less than the hypothesized population proportion of 0.44.
The alternative hypothesis sets the direction of the test, whether it is one-tailed or two-tailed, and helps in determining the critical values and the subsequent decisions.
Significance Level
The significance level, often denoted as \( \alpha \), is the probability threshold below which the null hypothesis will be rejected. It signifies the risk of making a Type I error.
In this scenario, a significance level of 0.02 or 2% is used for the critical-value approach, setting a stringent criterion for evidence against the null hypothesis.
The choice of significance level impacts the critical region of the test and is set by the researcher based on how much risk they are willing to take of falsely rejecting a true null hypothesis.
Type I Error
A Type I error occurs when the null hypothesis is rejected when it is actually true. This type of error is often called a 'false positive.'
The probability of making a Type I error is equal to the significance level \( \alpha \).
  • In this test, at a significance level of 0.02, the risk of a Type I error is 2%.
  • This implies there is a 2% chance that the conclusion from the test will incorrectly show a difference in population proportion when none exists.
The key is to choose a balance between the risk of Type I and Type II errors depending on the context and consequences.
P-value
The P-value is a measure used to determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining test results at least as extreme as the observed results, under the assumption that the null hypothesis is true.
It is computed from the z-score using a standard normal distribution table.
  • If the P-value is less than the chosen significance level \( \alpha \), the null hypothesis is rejected, indicating statistically significant results.
  • For instance, if \( \alpha = 0.01 \) and the computed P-value is lower, the evidence supports rejecting the null.
This metric provides an alternative perspective to the critical-value approach, summarizing data's evidence in a single value.
Z-score
The Z-score is a standardized value that tells you how many standard deviations an element is from the mean. In hypothesis testing, it is used to compare the sample statistic to the hypothesized parameter under the null hypothesis.
The formula calculates how far the sample proportion \( p \) deviates from the null hypothesis proportion \( P_0 \), adjusted by the standard error based on the sample size \( n \).
  • The calculated z-score is then compared to the critical z-value to make a decision on the null hypothesis.
  • In the exercise above, it determines whether the sample data aligns with or contradicts the null hypothesis. If the z-score is further into the critical region than the critical z-value, the null hypothesis is rejected.
The z-score acts as a bridge between the sample statistics and the theoretical probabilities of the standard normal distribution.

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Most popular questions from this chapter

Thirty percent of all people who are inoculated with the current vaccine that is used to prevent a disease contract the disease within a year. The developer of a new vaccine that is intended to prevent this disease wishes to test for significant evidence that the new vaccine is more effective. a. Determine the appropriate null and altemative hypotheses. b. The developer decides to study 100 randomly selected people by inoculating them with the new vaccine. If 84 or more of them do not contract the disease within a year, the developer will conclude that the new vaccine is superior to the old one. What significance level is the developer using for the test? c. Suppose 20 people inoculated with the new vaccine are studied and the new vaccine is concluded to be better than the old one if fewer than 3 people contract the disease within a year. What is the significance level of the test?

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=120 \text { versus } H_{1}: \mu>120 $$ A random sample of 81 observations taken from this population produced a sample mean of \(123.5 .\) The population standard deviation is known to be 15 . a. If this test is made at a \(2.5 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(a=.01 ?\) What if \(\alpha=.05\) ?

In Las Vegas, Nevada, and Atlantic City, New Jersey, tests are performed often on the various gaming devices used in casinos. For example, dice are often tested to determine if they are balanced. Suppose you are assigned the task of testing a die, using a two-tailed test to make sure that the probability of a 2 -spot is \(1 / 6\). Using a \(5 \%\) significance level, determine how many 2 -spots you would have to obtain to reject the null hypothesis when your sample size is a. 120 b. 1200 c. 12,000 Calculate the value of \(\hat{p}\) for each of these three cases. What can you say about the relationship between (1) the difference between \(\hat{p}\) and \(1 / 6\) that is necessary to reject the null hypothesis and (2) the sample size as it gets larger?

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.02\) and \(n=20\) b. A left-tailed test with \(\alpha=.01\) and \(n=16\) c. A right-tailed test with \(\alpha=.05\) and \(n=18\)

According to an estimate, 2 years ago the average age of all CEOs of medium- sized companies in the United States was 58 years. Jennifer wants to check if this is still true. She took a random sample of 70 such CEOs and found their mean age to be 55 years with a standard deviation of 6 years. a. Suppose that the probability of making a Type I error is selected to be zero. Can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? b. Using a \(1 \%\) significance level, can you conclude that the current mean age of all CEOs of medium-sized companies in the United States is different from 58 years? Use both approaches.
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