Textbook Question
Writing A null hypothesis is rejected with a level of significance of 0.10. Is it also rejected at a level of significance of 0.05? Explain.
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Ch. 7 - Hypothesis Testing with One Sample
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470
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Table of contents
- Ch. 1 - Introduction to Statistics87
- Ch. 2 - Descriptive Statistics176
- Ch. 3 - Probability184
- Ch. 4 - Discrete Probability Distributions102
- Ch. 5 - Normal Probability Distributions183
- Ch. 6 - Confidence Intervals154
- Ch. 7 - Hypothesis Testing with One Sample165
- Ch. 8 - Hypothesis Testing with Two Samples134
- Ch. 9 - Correlation and Regression87
- Ch. 10 - Chi-Square Tests and the F-Distribution87
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
Chapter 7, Problem 7.2.12
Graphical Analysis In Exercises 9–12, match the P-value or z-statistic with the graph that represents the corresponding area. Explain your reasoning.
z = -0.51
Verified step by step guidance1
Step 1: Understand the z-statistic. A z-statistic of -0.51 indicates that the value is 0.51 standard deviations below the mean in a standard normal distribution.
Step 2: Recall that the standard normal distribution is symmetric around the mean (z = 0). Negative z-values correspond to the left side of the distribution.
Step 3: Identify the shaded region in the graph. The blue area represents the cumulative probability to the left of z = -0.51.
Step 4: To find the P-value associated with z = -0.51, you would calculate the cumulative probability using a z-table or statistical software. This gives the proportion of the distribution that lies to the left of z = -0.51.
Step 5: Match the graph with the z-statistic. Since the graph shows the cumulative area to the left of z = -0.51, it correctly represents the P-value for this z-statistic.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Z-Statistic
The z-statistic is a measure that describes how many standard deviations a data point is from the mean of a distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. In hypothesis testing, the z-statistic helps determine the likelihood of observing a sample statistic under the null hypothesis.
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P-Value
The p-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence. In the context of the z-statistic, the p-value can be derived from the area under the normal distribution curve corresponding to the z-value.
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Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. It is symmetric around the mean, with about 68% of the data falling within one standard deviation. Understanding the properties of the normal distribution is crucial for interpreting z-statistics and p-values, as many statistical tests assume normality.
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Related Practice
Textbook Question
In Exercises 3–6, determine whether a normal sampling distribution can be used. If it can be used, test the claim.
Claim: p > 0.70, α=0.04. Sample statistics: p_hat = 0.64, n=225
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Textbook Question
Hypothesis Testing Using Rejection Regions In Exercises 23–30, (a) identify the claim and state H0 and Ha, (b) find the critical value(s) and identify the rejection region(s), (c) find the standardized test statistic X^2, (d) decide whether to reject or fail to reject the null hypothesis, and (e) interpret the decision in the context of the original claim. Assume the population is normally distributed.
Salaries The annual salaries (in dollars) of 15 randomly chosen senior level graphic design specialists are shown in the table at the left. At α=0.05, is there enough evidence to support the claim that the standard deviation of the annual salaries is different from \$13,056?
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