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Ch. 6 - Confidence Intervals
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 6 - Confidence IntervalsProblem 6.R.23
Chapter 6, Problem 6.R.23

In Exercise 19, would it be unusual for the population proportion to be 38%? Explain.

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1
Step 1: Identify the given population proportion (p) from the problem. Here, the population proportion is 38%, which can be written as p = 0.38.
Step 2: Determine the context of 'unusual' in statistics. Typically, an event is considered unusual if it lies more than 2 standard deviations away from the mean in a normal distribution.
Step 3: Calculate the standard error (SE) of the population proportion using the formula: SE = sqrt((p(1-p))/n), where p is the population proportion and n is the sample size. If the sample size (n) is not provided, it must be specified to proceed.
Step 4: Use the calculated standard error to determine the range of usual values. The range is given by: p ± 2 × SE. This range represents the interval within which values are considered usual.
Step 5: Compare the given population proportion (38%) to the calculated range. If 38% lies outside this range, it would be considered unusual. Otherwise, it is not unusual.

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Key Concepts

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

Population Proportion

Population proportion refers to the fraction of a population that possesses a certain characteristic. It is a key parameter in statistics, often denoted as 'p', and is used to make inferences about the entire population based on sample data. Understanding the population proportion helps in determining the likelihood of observing certain outcomes in a sample.
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Statistical Significance

Statistical significance assesses whether the observed data deviates from what would be expected under a null hypothesis. In the context of population proportions, a proportion of 38% may be evaluated against a hypothesized value to determine if it is statistically unusual. This involves hypothesis testing and calculating p-values to draw conclusions about the population.
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Confidence Intervals

A confidence interval provides a range of values within which the true population proportion is likely to fall, based on sample data. It reflects the uncertainty associated with estimating the population parameter. If the 38% proportion lies outside the confidence interval, it may suggest that this value is unusual or not representative of the population.
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Related Practice
Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.98, s = 0.9, n = 12, xbar = 6.8

Textbook Question

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 912 U.S. adults in Generation Z (born after 1996), 383 said they are at least somewhat likely to consider an electric vehicle for their next vehicle purchase. (Adapted from Pew Research Center)

Textbook Question

You wish to estimate, with 95% confidence, the population proportion of U.S. adults who have taken or planned to take a winter vacation in a recent year. Your estimate must be accurate within 5% of the population proportion.

a. No preliminary estimate is available. Find the minimum sample size needed.

Textbook Question

In Exercises 27–30, find the critical values and for the level of confidence c and sample size n.

c = 0.99, n = 10

Textbook Question

In Exercises 19–22, let p be the population proportion for the situation. (a) Find point estimates of p and q, (b) construct 90% and 95% confidence intervals for p, and (c) interpret the results of part (b) and compare the widths of the confidence intervals.

In a survey of 73,901 college graduates, 23,991 obtained a postgraduate degree. (Adapted from Gallup)

Textbook Question

In Exercises 13–16, (a) find the margin of error for the values of c, s, and n, and (b) construct the confidence interval for using the t-distribution. Assume the population is normally distributed.

c = 0.90, s = 25.6, n = 16, xbar = 72.1