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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Ch. 8 - Hypothesis Testing
Triola14th EditionElementary StatisticsISBN: 9780137366446
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Table of contents
- Ch. 1 - Introduction to Statistics88
- Ch. 2 - Exploring Data with Tables and Graphs70
- Ch. 3 - Describing, Exploring, and Comparing Data66
- Ch. 4 - Probability112
- Ch. 5 - Discrete Probability Distributions109
- Ch. 6 - Normal Probability Distributions122
- Ch. 7 - Estimating Parameters and Determining Sample Sizes107
- Ch. 8 - Hypothesis Testing75
- Ch. 9 - Inferences from Two Samples94
- Ch. 10 - Correlation and Regression80
- Ch. 11 - Goodness-of-Fit and Contingency Tables29
- Ch. 12 - Analysis of Variance42
Chapter 8, Problem 8.5.3b
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.
b. For random samples of size 860, what sample proportions of male births are at least as extreme as the sample proportion of 426/860?
Verified step by step guidance1
Calculate the sample proportion of male births using the formula: . This will give the observed sample proportion.
Determine the null hypothesis proportion, which is given as 0.512. This represents the assumed proportion of male births in the population under the null hypothesis.
Compute the standard error (SE) of the sample proportion using the formula: , where is the null hypothesis proportion (0.512) and is the sample size (860).
Calculate the z-score for the observed sample proportion using the formula: , where is the observed sample proportion, is the null hypothesis proportion, and is the standard error.
Find the z-scores that correspond to sample proportions at least as extreme as the observed proportion. This involves considering both tails of the distribution (greater than the observed proportion and less than the symmetric lower proportion). Use a z-table or statistical software to determine the corresponding probabilities.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Proportion
The sample proportion is the ratio of the number of successes (in this case, male births) to the total number of observations in the sample. It is calculated by dividing the number of boys (426) by the total number of births (860), yielding a sample proportion of 0.495. This value is crucial for hypothesis testing as it serves as the basis for comparing against the hypothesized population proportion.
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Sampling Distribution of Sample Proportion
Hypothesis Testing
Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. In this context, the null hypothesis states that the proportion of male births is equal to 0.512. The test will evaluate if the observed sample proportion is significantly different from this hypothesized value, using a significance level to make a decision.
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Guided course
06:21Step 1: Write Hypotheses
At Least As Extreme
The phrase 'at least as extreme' refers to the values of the sample proportion that are as far away from the hypothesized population proportion as the observed sample proportion. This concept is used to determine the critical values for the hypothesis test, which helps in assessing whether the observed sample proportion is statistically significant compared to the expected proportion under the null hypothesis.
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04:48Comparing Mean vs. Median
Related Practice
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
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
b. Find the value of the test statistic.
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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.
b. Use the P-value method with a 0.05 significance level to test the claim that the proportion of zeros equals 0.1.
133
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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.
Number and Proportions
b. Identify the sample proportion and use the symbol that represents it.
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