Textbook Question
In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?
c=0.90, x̅=8.21, σ=0.62, n=8
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Ch. 8 - Hypothesis Testing with Two Samples
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 8, Problem 8.CR.4
In Exercises 3–6, construct the indicated confidence interval for the population mean . Which distribution did you use to create the confidence interval?
c=0.95, x̅=3.46, s=1.63, n=16
Verified step by step guidance1
Step 1: Identify the type of distribution to use. Since the sample size (n = 16) is small and the population standard deviation is not provided, use the t-distribution to construct the confidence interval.
Step 2: Determine the degrees of freedom (df) for the t-distribution. The formula for degrees of freedom is df = n - 1. Here, df = 16 - 1 = 15.
Step 3: Find the critical t-value (t*) for a 95% confidence level and 15 degrees of freedom. Use a t-distribution table or statistical software to find this value.
Step 4: Calculate the margin of error (ME) using the formula: ME = t* × (s / √n), where s is the sample standard deviation, and n is the sample size.
Step 5: Construct the confidence interval using the formula: Confidence Interval = x̅ ± ME, where x̅ is the sample mean. This will give you the lower and upper bounds of the interval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Confidence Interval
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. For example, a 95% confidence interval suggests that if we were to take many samples and construct intervals in the same way, approximately 95% of those intervals would contain the true population mean.
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Introduction to Confidence Intervals
Normal Distribution
The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, defined by its mean and standard deviation. When constructing confidence intervals for the population mean, if the sample size is large (typically n > 30) or the population is normally distributed, the sample mean can be assumed to follow a normal distribution, allowing for the use of z-scores.
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06:23Using the Normal Distribution to Approximate Binomial Probabilities
t-Distribution
The t-distribution is a type of probability distribution that is used when the sample size is small (n < 30) and the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails, which provides a more accurate estimate of the confidence interval for the population mean in such cases, as it accounts for the increased variability in smaller samples.
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Related Practice
Textbook Question
Constructing Confidence Intervals for p1-p2 You can construct a confidence interval for the difference between two population proportions p1-p2 by using the inequality below.
[Image] Complicated mathematical formula.
In Exercises 23–26, construct the indicated confidence interval for p1-p2. Assume the samples are random and independent.
Critical Threats Repeat Exercise 25 but with a 99% confidence interval. Describe the likelihood that equal proportions of the population see cyberterrorism and the spread of infectious diseases as critical threats in the next 10 years.
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Textbook Question
The mean room rate for two adults for a random sample of 26 three-star hotels in Cincinnati has a sample standard deviation of \$31. Assume the population is normally distributed. (Adapted from Expedia)
Construct a 99% confidence interval for the population variance.
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Textbook Question
An education organization claims that the mean SAT scores for male athletes and male non-athletes at a college are different. A random sample of 26 male athletes at the college has a mean SAT score of 1189 and a standard deviation of 218. A random sample of 18 male non-athletes at the college has a mean SAT score of 1376 and a standard deviation of 186. At α=0.05, can you support the organization’s claim? Interpret the decision in the context of the original claim. Assume the populations are normally distributed and the population variances are equal.
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Textbook Question
In a survey of 4860 U.S. adults, 77% said they would date or have already dated someone whose religion was different from theirs. (Source: Pew Research Center)
Construct a 95% confidence interval for the proportion of U.S. adults who say they would date or have already dated someone whose religion was different from theirs.
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Textbook Question
[APPLET] The annual earnings (in dollars) for 30 randomly selected locksmiths are shown below. Assume the population is normally distributed. (Adapted from Salary.com)
48,69446,85642,91261,67271,11254,861
69,45471,84159,75169,61254,28452,166
66,36048,16465,27235,25061,12765,397
58,92558,91659,01753,07045,19969,941
69,49257,08553,82952,69268,29853,792
A researcher claims that the mean annual earnings for locksmiths is \$55,000. At α=0.05, can you reject the researcher’s claim? Interpret the decision in the context of the original claim.
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