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Chapter 10: Problem 2

True or False: When testing a hypothesis using the Classical Approach, if the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reiect the null hypothesis.

Short Answer

Expert verified
True.

Step by step solution

01

- Understand the Hypothesis Testing

Hypothesis testing involves comparing a sample proportion to a hypothetical proportion stated in the null hypothesis. We decide whether to reject the null hypothesis based on the sample proportion's deviation from this hypothetical proportion.
02

- Define 'Too Many Standard Deviations'

In hypothesis testing, 'too many standard deviations' typically means that the sample proportion lies beyond a certain critical value. This critical value is determined based on the significance level (e.g., 0.05).
03

- Compare Sample Proportion to Critical Value

Calculate the test statistic, which measures how many standard deviations the sample proportion is from the proportion stated in the null hypothesis. If this test statistic exceeds the critical value, it is considered 'too many standard deviations.'
04

- Decision Rule

If the test statistic indicates that the sample proportion is 'too many standard deviations' (i.e., beyond the critical value), we reject the null hypothesis. If it is within the acceptable range, we do not reject the null hypothesis.
05

- Answer the Question

Based on the Classical Approach to hypothesis testing, the statement is true. If the sample proportion is too many standard deviations from the proportion stated in the null hypothesis, we reject the null hypothesis.

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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 is a fundamental concept in hypothesis testing. It represents a statement or default position that there is no effect or no difference. For example, if you're testing a new drug, the null hypothesis might be that the drug has no effect on patients. In hypothesis testing, we aim to gather evidence against the null hypothesis, rather than to prove it directly. If our sample data shows a significant deviation from what we would expect under the null hypothesis, we may reject the null hypothesis in favor of an alternative hypothesis.
sample proportion
When we talk about the sample proportion, we're referring to the proportion of a particular outcome within a sample of data. For example, if you surveyed 100 people and 40 of them preferred a certain product, the sample proportion would be 0.40. This sample proportion is critical in hypothesis testing because it allows us to make inferences about the overall population proportion. We compare the sample proportion to a hypothetical proportion stated in the null hypothesis to determine if there is evidence to reject the null hypothesis.
standard deviations
Standard deviations measure the amount of variability or dispersion in a set of data. In the context of hypothesis testing, standard deviations help quantify how far the sample proportion is from the hypothetical proportion stated in the null hypothesis. When we calculate a test statistic in hypothesis testing, we're essentially measuring how many standard deviations the sample proportion is from the null hypothesis proportion. If the test statistic is too many standard deviations, it suggests that the sample proportion is significantly different from the null hypothesis proportion.
critical value
The critical value is a key concept in hypothesis testing. It represents a threshold that the test statistic must exceed in order for us to reject the null hypothesis. The critical value is determined based on the chosen significance level (typically denoted by alpha, such as 0.05). It marks the boundary of the acceptable range of values that we would expect to see if the null hypothesis is true. If the test statistic exceeds the critical value (meaning it falls in the rejection region), we have enough evidence to reject the null hypothesis.
significance level
The significance level, often denoted by alpha (伪), is the probability of rejecting the null hypothesis when it is actually true. This is also known as the Type I error rate. A common significance level used in hypothesis testing is 0.05, though other levels such as 0.01 or 0.10 may also be used depending on the context. The significance level determines the critical value; for example, a 0.05 significance level implies that there is a 5% chance of rejecting the null hypothesis when it is true. Choosing an appropriate significance level is crucial for balancing the risks of Type I and Type II errors in the hypothesis test.

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Most popular questions from this chapter

According to the Centers for Disease Control, \(15.2 \%\) of American adults experience migraine headaches. Stress is a major contributor to the frequency and intensity of headaches. A massage therapist feels that she has a technique that can reduce the frequency and intensity of migraine headaches. (a) Determine the null and alternative hypotheses that would be used to test the effectiveness of the massage therapist's techniques. (b) A sample of 500 American adults who participated in the massage therapist's program results in data that indicate that the null hypothesis should be rejected. Provide a statement that supports the massage therapist's program. (c) Suppose, in fact, that the percentage of patients in the program who experience migraine headaches is \(15.3 \%\). Was a Type I or Type II error committed?

Simulate drawing 100 simple random samples of size \(n=40\) from a population whose proportion is 0.3 (a) Test the null hypothesis \(H_{0}: p=0.3\) versus \(H_{1}: p \neq 0.3\) for each simulated sample. (b) If we test the hypothesis at the \(\alpha=0.1\) level of significance, how many of the 100 samples would you expect to result in a Type I error? (c) Count the number of samples that lead to a rejection of the null hypothesis. Is it close to the expected value determined in part (b)? (d) How do we know that a rejection of the null hypothesis results in making a Type I error in this situation?

A simple random sample of size \(n=20\) is drawn from a population that is normally distributed. The sample variance is found to be \(49.3 .\) Determine whether the population variance is less than 95 at the \(\alpha=0.1\) level of significance.

NCAA rules require the circumference of a softball to be \(12 \pm 0.125\) inches. Suppose that the NCAA also requires that the standard deviation of the softball circumferences not exceed 0.05 inch. A representative from the NCAA believes the manufacturer does not meet this requirement. She collects a random sample of 20 softballs from the production line and finds that \(s=0.09\) inch. Is there enough evidence to support the representative's belief at the \(\alpha=0.05\) level of significance?

Study Time Go to www.pearsonhighered.com/sullivanstats to obtain the data file \(10_{-} 3_{-} 26\) using the file format of your choice for the version of the text you are using. The data represent the amount of time students in Sullivan's online statistics course spent studying for Section 4.1 -Scatter Diagrams and Correlation. (a) Draw a histogram of the data. Describe the shape of the distribution. (b) Draw a boxplot of the data. Are there any outliers? (c) Based on the shape of the histogram and boxplot, explain why a large sample size is necessary to perform inference on the mean using the normal model. (d) According to MyStatLab, the mean time students would spend on this assignment nationwide is 95 minutes. Treat the data as a random sample of all Sullivan online statistics students. Do the sample data suggest that Sullivan's students are any different from the country as far as time spent on Section 4.1 goes? Use an \(\alpha=0.05\) level of significance.
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