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Chapter 10: Problem 41

Suppose we are testing the hypothesis \(H_{0}: p=0.3\) versus \(H_{1}: p>0.3\) and we find the \(P\) -value to be \(0.23 .\) Explain what this means. Would you reject the null hypothesis? Why?

Short Answer

Expert verified
Do not reject the null hypothesis because the P-value of 0.23 is greater than the significance level of 0.05.

Step by step solution

01

Understand the Hypotheses

The null hypothesis (H_0) states that the population proportion ( p) is 0.3 and the alternative hypothesis ( H_1) states that the population proportion ( p) is greater than 0.3.
02

Identify the P-value

The P-value for the test is given as 0.23. The P-value measures the probability of obtaining a test statistic at least as extreme as the one obtained, assuming the null hypothesis is true.
03

Compare the P-value to the Significance Level

Typically, we compare the P-value to a significance level ( alpha ), often set at 0.05. If P-value is less than or equal to alpha , we reject the null hypothesis.
04

Make a Decision

Since 0.23 is greater than 0.05, we do NOT reject the null hypothesis. The P-value is not small enough to provide strong evidence against H_0 .

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Key Concepts

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

P-value interpretation
The P-value plays a crucial role in hypothesis testing. It indicates the probability of obtaining a test statistic at least as extreme as the one observed, given that the null hypothesis is true. For example, a P-value of 0.23 means there is a 23% chance of observing the given or a more extreme result under the assumption that the null hypothesis holds. A higher P-value, like 0.23, suggests that the observed result is relatively likely when the null hypothesis is true.
null hypothesis
The null hypothesis, usually denoted as H_0, is a statement that there is no effect or no difference. It serves as the starting point for testing. In our example, the null hypothesis is that the population proportion (p) is 0.3. This hypothesis remains 'innocent until proven guilty' and is only rejected if the data provides strong enough evidence against it. We assume H_0 is true when calculating probabilities like the P-value.
significance level
The significance level, denoted by 伪 (alpha), is the threshold that determines when we reject the null hypothesis. Commonly, 伪 is set at 0.05, which corresponds to a 5% risk of rejecting the null hypothesis when it is actually true (Type I error). If the P-value is less than or equal to 伪, we reject the null hypothesis. For example, if 伪 = 0.05 and the observed P-value is 0.23, we do not reject H_0 because 0.23 > 0.05.
alternative hypothesis
The alternative hypothesis, denoted as H_1, is what you want to support. It usually represents a new effect or difference. In our example, H_1 states that the population proportion (p) is greater than 0.3. While the null hypothesis assumes no change or effect, the alternative hypothesis suggests a specific direction or magnitude of change. If evidence strong enough to reject H_0 is found, we provide support for H_1. However, a high P-value (like 0.23) does not offer such strong evidence, so we do not reject H_0.

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Most popular questions from this chapter

True or False: Sample evidence can prove a null hypothesis is true.

In \(2000,58 \%\) of females aged 15 and older lived alone, according to the U.S. Census Bureau. A sociologist tests whether this percentage is different today by conducting a random sample of 500 females aged 15 and older and finds that 285 are living alone. Is there sufficient evidence at the \(\alpha=0.1\) level of significance to conclude the proportion has changed since \(2000 ?\)

The website pundittracker.com keeps track of predictions made by individuals in finance, politics, sports, and entertainment. Jim Cramer is a famous TV financial personality and author. Pundittracker monitored 678 of his stock predictions (such as a recommendation to buy the stock) and found that 320 were correct predictions. Treat these 678 predictions as a random sample of all of Cramer's predictions. (a) Determine the sample proportion of predictions Cramer got correct. (b) Suppose that we want to know whether the evidence suggests Cramer is correct less than half the time. State the null and alternative hypotheses. (c) Verify the normal model may be used to determine the \(P\) -value for this hypothesis test. (d) Draw a normal model with area representing the \(P\) -value shaded for this hypothesis test. (e) Determine the \(P\) -value based on the model from part (d). (f) Interpret the \(P\) -value. (g) Based on the \(P\) -value, what does the sample evidence suggest? That is, what is the conclusion of the hypothesis test? Assume an \(\alpha=0.05\) level of significance.

In \(1994,52 \%\) of parents with children in high school felt it was a serious problem that high school students were not being taught enough math and science. A recent survey found that 256 of 800 parents with children in high school felt it was a serious problem that high school students were not being taught enough math and science. Do parents feel differently today than they did in 1994 ? Use the \(\alpha=0.05\) level of significance? Source: Based on "Reality Check: Are Parents and Students Ready for More Math and Science?" Public Agenda, \(2006 .\)

A machine fills bottles with 64 fluid ounces of liquid. The quality-control manager determines that the fill levels are normally distributed with a mean of 64 ounces and a standard deviation of 0.42 ounce. He has an engineer recalibrate the machine in an attempt to lower the standard deviation. After the recalibration, the quality-control manager randomly selects 19 bottles from the line and determines that the standard deviation is 0.38 ounce. Is there less variability in the filling machine? Use the \(\alpha=0.01\) level of significance.
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