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Ch. 7 - Estimating Parameters and Determining Sample Sizes
Triola - Elementary Statistics 14th Edition
Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook
All textbooksTriola 14th EditionCh. 7 - Estimating Parameters and Determining Sample SizesProblem 7.1.7
Chapter 7, Problem 7.1.7

Finding Critical Values


In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.


99.5%

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1
Step 1: Understand the problem. The critical value z_{α/2} is a z-score that corresponds to the given confidence level. For a 99.5% confidence level, the area in the middle of the standard normal distribution is 0.995, and the remaining area (α) is split equally between the two tails.
Step 2: Calculate α. Subtract the confidence level from 1 to find the total area in the tails: α = 1 - 0.995 = 0.005.
Step 3: Divide α by 2 to find the area in one tail: α/2 = 0.005 / 2 = 0.0025. This represents the area in the left tail of the standard normal distribution.
Step 4: Use a z-table or statistical software to find the z-score that corresponds to the cumulative area of 1 - α/2 = 1 - 0.0025 = 0.9975. This cumulative area represents the area to the left of the critical value z_{α/2}.
Step 5: Interpret the result. The z-score you find is the critical value z_{α/2} for the 99.5% confidence level. This value is symmetric, so the critical values are ±z_{α/2}.

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Key Concepts

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

Critical Value

A critical value is a point on the scale of the test statistic that separates the region where the null hypothesis is rejected from the region where it is not rejected. In the context of confidence intervals, it corresponds to the z-score that captures the desired level of confidence, indicating how many standard deviations away from the mean a data point must be to fall within a certain confidence level.
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Confidence Level

The confidence level represents the probability that the confidence interval will contain the true population parameter. It is expressed as a percentage, such as 99.5%, indicating that if we were to take many samples and build a confidence interval from each, approximately 99.5% of those intervals would contain the true parameter. Higher confidence levels result in wider intervals.
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Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations. In the context of finding critical values, the z-score corresponding to a specific confidence level helps determine the cutoff points for the tails of the normal distribution, which are essential for constructing confidence intervals.
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Related Practice
Textbook Question

Second-Hand Smoke Refer to Data Set 15 “Passive and Active Smoke” and construct a 95% confidence interval estimates of the mean cotinine level in each of three samples: (1) people who smoke; (2) people who don’t smoke but are exposed to tobacco smoke at home or work; (3) people who don’t smoke and are not exposed to smoke. Measuring cotinine in people’s blood is the most reliable way to determine exposure to nicotine. What do the confidence intervals suggest about the effects of smoking and second-hand smoke?

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Textbook Question

Heights of Presidents Refer to Data Set 22 “Presidents” in Appendix B. Treat the data as a sample and find the proportion of presidents who were taller than their opponents. Use that result to construct a 95% confidence interval estimate of the population percentage. Based on the result, does it appear that greater height is an advantage for presidential candidates? Why or why not?

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Textbook Question

Seating Choice In a 3M Privacy Filters poll, respondents were asked to identify their favorite seat when they fly, and the results include these responses: window, window, other, other. Letting “window” and letting “other”, those four responses can be represented as {1, 1, 0, 0}. Here are ten bootstrap samples for those responses: [Image]

Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the proportion of respondents who indicated their favorite seat is “window.”

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Textbook Question

Professor Evaluation Scores Listed below are student evaluation scores of professors from Data Set 28 “Course Evaluations” in Appendix B. Construct a 95% confidence interval estimate of for each of the two data sets. Does there appear to be a difference in variation?

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Textbook Question

Finding Critical Values.


In Exercises 5–8, find the critical value z=a/2 that corresponds to the given confidence level.


90%

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Textbook Question

Mean IQ of Data Scientists See the preceding exercise, in which we can assume that sigma=15 for the IQ scores. Data scientists are a group with IQ scores that vary less than the IQ scores of the general population. Find the sample size needed to estimate the mean IQ of data scientists, given that we want 98% confidence that the sample mean is within 2 IQ points of the population mean. Does the sample size appear to be practical?

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