6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
6:00 minutesProblem 6.6.18c
Textbook Question
Sleepwalking Assume that 29.2% of people have sleepwalked (based on “Prevalence and Comorbidity of Nocturnal Wandering in the U.S. Adult General Population, by Ohayon et al., Neurology, Vol. 78, No. 20). Assume that in a random sample of 1480 adults, 455 have sleepwalked.
c. What does the result suggest about the rate of 29.2%?
Verified step by step guidance1
Step 1: Define the null hypothesis (Hâ‚€) and the alternative hypothesis (Hâ‚). The null hypothesis states that the proportion of people who have sleepwalked is 29.2% (p = 0.292). The alternative hypothesis states that the proportion is different from 29.2% (p ≠0.292).
Step 2: Calculate the sample proportion (±èÌ‚). The sample proportion is given by the formula: , where x is the number of people who have sleepwalked (455) and n is the sample size (1480).
Step 3: Compute the standard error (SE) of the sample proportion. The formula for the standard error is: , where p is the hypothesized proportion (0.292) and n is the sample size (1480).
Step 4: Calculate the test statistic (z-score). The formula for the z-score is: , where ±èÌ‚ is the sample proportion, p is the hypothesized proportion, and SE is the standard error.
Step 5: Compare the calculated z-score to the critical z-value for the chosen significance level (e.g., α = 0.05 for a two-tailed test). If the z-score falls outside the range of the critical z-values, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the result in the context of whether the sample proportion suggests a significant difference from the hypothesized rate of 29.2%.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sampling Distribution
The sampling distribution refers to the probability distribution of a statistic (like a sample proportion) obtained from a large number of samples drawn from a specific population. It helps in understanding how sample statistics vary and is crucial for making inferences about the population based on sample data.
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Sampling Distribution of Sample Proportion
Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (e.g., the population proportion is 29.2%) and an alternative hypothesis, then using sample data to determine whether to reject the null hypothesis based on a significance level.
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Confidence Intervals
A confidence interval is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence (e.g., 95%). It provides an estimate of the uncertainty around the sample proportion and helps assess whether the observed sample proportion significantly deviates from the hypothesized population proportion.
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