Ch. 8 - Hypothesis Testing

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

Ch. 8 - Hypothesis Testing

Problem 8.5.3a

Chapter 8, Problem 8.5.3a

At Least As Extreme A random sample of 860 births in New York State included 426 boys, and that sample is to be used for a test of the common belief that the proportion of male births in the population is equal to 0.512.

a. In testing the common belief that the proportion of male babies is equal to 0.512, identify the values of p^ and p.

Verified step by step guidance

Step 1: Understand the problem. The goal is to test the common belief that the proportion of male births in the population is equal to 0.512. Here, 'p' represents the hypothesized population proportion, and 'p^' (read as 'p-hat') represents the sample proportion.

Step 2: Identify the value of 'p'. The problem states that the common belief is that the proportion of male births in the population is 0.512. Therefore, 'p' = 0.512.

Step 3: Calculate the sample proportion 'p^'. The formula for 'p^' is:

p^= \frac{x}{n}

, where 'x' is the number of successes (male births) and 'n' is the total sample size. Here, x = 426 and n = 860.

Step 4: Substitute the values into the formula for 'p^'. Using the formula:

p^= \frac{426}{860}

. This will give the sample proportion of male births.

Step 5: Interpret the values. 'p' is the hypothesized proportion (0.512), and 'p^' is the sample proportion calculated from the data. These values will be used in further statistical tests, such as a hypothesis test for proportions, to determine if the observed sample proportion significantly differs from the hypothesized population proportion.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Sample Proportion (p^)

The sample proportion, denoted as p^ (p-hat), is the ratio of the number of successes (in this case, the number of boys born) to the total number of observations in the sample. It provides an estimate of the population proportion based on the sample data. For the given question, p^ can be calculated as 426 boys out of 860 total births, which helps in assessing the validity of the common belief about the population proportion.

Population Proportion (p)

The population proportion, denoted as p, represents the true proportion of a certain characteristic in the entire population. In this scenario, it is the hypothesized proportion of male births, which is stated to be 0.512. Understanding the difference between p and p^ is crucial for hypothesis testing, as it allows us to compare the sample data against the established belief about the population.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to support a specific claim about a population parameter. In this context, the null hypothesis (H0) would state that the population proportion of male births is equal to 0.512, while the alternative hypothesis (H1) would suggest that it is not. This process involves calculating test statistics and p-values to make informed decisions regarding the hypotheses.

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Guided course

06:21

Step 1: Write Hypotheses

Related Practice

Textbook Question

Claim of “At Least” or “At Most”

How do the following results change?

a. Chapter Problem claim is changed to this: “At least 50% of Internet users utilize two-factor authentication to protect their online data.”

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Textbook Question

Identifying H0 and H1

In Exercises 5–8, do the following:

a. Express the original claim in symbolic form.

b. Identify the null and alternative hypotheses.

Systolic Blood Pressure Claim: Healthy adults have systolic blood pressure levels with a standard deviation greater than 5 mm Hg. Sample data: Data Set 1 “Body Data” in Appendix B shows that for 300 healthy adults, the systolic blood pressure amounts have a standard deviation of 15.85 mm Hg.

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Textbook Question

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

a. Use the critical value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.

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Textbook Question

Statistical Literacy and Critical Thinking

In Exercises 1–4, use the results from a Hankook Tire Gauge Index survey of a simple random sample of 1020 adults. Among the 1020 respondents, 86% rated themselves as above average drivers. We want to test the claim that more than 3/4 of adults rate themselves as above average drivers.

Null and Alternative Hypotheses and Test Statistic

a. Identify the null hypothesis and the alternative hypothesis.

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Textbook Question

Finding Critical Values

In Exercises 17–20, refer to the information in the given exercise and use a 0.05 significance level for the following.

a. Find the critical value(s).

b. Should we reject H0 or should we fail to reject H0?

Exercise 15

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Textbook Question

RESAMPLING

a. In general, what does it mean to “resample” the following data set consisting of wait times (minutes) of customers waiting in line for the Space Mountain ride at Walt Disney World: 50, 25, 75, 35, 50?

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