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Ch. 6 - Normal Probability Distributions
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 6 - Normal Probability DistributionsProblem 6.6.16b
Chapter 6, Problem 6.6.16b

Eye Color Based on a study by Dr. P. Sorita at Indiana University, assume that 12% of us have green eyes. In a study of 650 people, it is found that 86 of them have green eyes.


b. Is 86 people with green eyes significantly high?

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Step 1: Define the problem in terms of probability. We are tasked with determining if 86 people with green eyes out of 650 is significantly high, given that the population proportion of green eyes is 12% (p = 0.12). This involves using the concept of a sampling distribution for proportions.
Step 2: Calculate the mean of the sampling distribution for the proportion of green eyes. The mean is given by \( \mu_p = p \), where \( p = 0.12 \).
Step 3: Calculate the standard deviation of the sampling distribution for the proportion. The formula is \( \sigma_p = \sqrt{\frac{p(1-p)}{n}} \), where \( p = 0.12 \), \( 1-p = 0.88 \), and \( n = 650 \).
Step 4: Convert the observed number of green-eyed people (86) into a sample proportion \( \hat{p} \). The formula is \( \hat{p} = \frac{x}{n} \), where \( x = 86 \) and \( n = 650 \).
Step 5: Compute the z-score to determine how many standard deviations \( \hat{p} \) is from \( \mu_p \). The formula is \( z = \frac{\hat{p} - \mu_p}{\sigma_p} \). Compare the z-score to a critical value (e.g., \( z = 1.96 \) for a 5% significance level) to decide if 86 is significantly high.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to reject a null hypothesis. In this context, the null hypothesis might state that the proportion of people with green eyes in the sample is equal to the expected proportion of 12%. By comparing the observed number of individuals with green eyes to what is expected, we can assess whether the difference is statistically significant.
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Step 1: Write Hypotheses

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a result is statistically significant. Commonly set at 0.05, it represents a 5% risk of concluding that a difference exists when there is none. In this scenario, if the proportion of green-eyed individuals in the sample significantly exceeds the expected 12%, we would reject the null hypothesis at the chosen significance level.
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Step 4: State Conclusion Example 4

Proportion and Sample Size

Proportion refers to the fraction of a population that exhibits a certain characteristic, such as having green eyes. In this case, we compare the observed proportion of green-eyed individuals (86 out of 650) to the expected proportion of 12%. The sample size is crucial because larger samples tend to provide more reliable estimates of population parameters, affecting the power of the hypothesis test.
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Sampling Distribution of Sample Proportion
Related Practice
Textbook Question

In Exercises 1 and 2, use the following wait times (minutes) at 10:00 AM for the Tower of Terror ride at Disney World (from Data Set 33 “Disney World Wait Times” in Appendix B).


35 35 20 50 95 75 45 50 30 35 30 30


c. Find the standard deviation s.

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Textbook Question

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Women’s heights are normally distributed with mean 63.7 in. and standard deviation 2.9 in.

If the Navy changes the height requirements so that all women are eligible except the shortest 3% and the tallest 3%, what are the new height requirements for women?

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Textbook Question

Hershey Kisses Based on Data Set 38 “Candies” in Appendix B, weights of the chocolate in Hershey Kisses are normally distributed with a mean of 4.5338 g and a standard deviation of 0.1039 g


b. What is the value of the median?

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Textbook Question

College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.


b. What value do the sample means target? That is, what is the mean of all such sample means?

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Textbook Question

Transformations The heights (in inches) of women listed in Data Set 1 “Body Data” in Appendix B have a distribution that is approximately normal, so it appears that those heights are from a normally distributed population.


b. If each height is converted from inches to centimeters, are the heights in centimeters also normally distributed?

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Textbook Question

Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.


b. Find the mean of the sampling distribution.

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