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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 7.4.12b
Chapter 7, Problem 7.4.12b

Minting Quarters Listed below are weights (grams) of quarters minted after 1964 (based on Data Set 40 “Coin Weights” in Appendix B).


b. Specifications require that the quarters have a weight of 5.670 g. What does the confidence interval suggest about that specification?


Weights of quarters: 5.7790, 5.5928, 5.6486, 5.6661, 5.5491, 5.7239, 5.5591, 5.5864, 5.6872, 5.6274 grams.

Verified step by step guidance
1
Step 1: Calculate the sample mean (x̄) of the given weights. Use the formula for the mean: x̄ = (Σx) / n, where Σx is the sum of all weights and n is the number of weights.
Step 2: Calculate the sample standard deviation (s) using the formula: s = sqrt((Σ(x - x̄)^2) / (n - 1)), where x̄ is the sample mean, x represents each individual weight, and n is the sample size.
Step 3: Determine the confidence interval for the mean weight using the formula: CI = x̄ ± (t * (s / sqrt(n))), where t is the critical value from the t-distribution for the desired confidence level (e.g., 95%) and degrees of freedom (df = n - 1).
Step 4: Compare the confidence interval to the specification weight of 5.670 g. If the specification weight falls within the confidence interval, it suggests that the quarters meet the specification. If it falls outside, it suggests potential issues with meeting the specification.
Step 5: Interpret the results in the context of the problem. Discuss whether the confidence interval supports the claim that the quarters meet the specification weight of 5.670 g.

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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 true population parameter. It provides an estimate of uncertainty around a sample mean, indicating how much the sample mean might vary from the actual population mean. For example, if a 95% confidence interval for the mean weight of quarters is (5.65 g, 5.70 g), we can be 95% confident that the true mean weight lies within this range.
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Introduction to Confidence Intervals

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 mean weight of quarters is 5.670 g) and an alternative hypothesis. By analyzing the sample data, we can determine whether to reject or fail to reject the null hypothesis, which helps assess if the sample provides enough evidence to support a claim about the population.
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Step 1: Write Hypotheses

Sample Mean and Standard Deviation

The sample mean is the average of a set of observations, calculated by summing the values and dividing by the number of observations. The standard deviation measures the dispersion of the sample data around the mean, indicating how spread out the values are. In the context of the quarter weights, calculating the sample mean and standard deviation is essential for constructing the confidence interval and understanding the variability in the weights.
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Related Practice
Textbook Question

Mint Specs Listed below are weights (grams) from a simple random sample of pennies produced after 1983 (from Data Set 40 “Coin Weights” in Appendix B).


b. How does the result compare to the confidence interval found in Exercise 14 in Section 7-3?


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Textbook Question

Mean Pulse Rate of Females Data Set 1 “Body Data” in Appendix B includes pulse rates of 147 randomly selected adult females, and those pulse rates vary from a low of 36 bpm to a high of 104 bpm. Find the minimum sample size required to estimate the mean pulse rate of adult females. Assume that we want 99% confidence that the sample mean is within 2 bpm of the population mean.


b. Assume that sigma=12.5 bpm, based on the value of s=12.5 bpm for the sample of 147 female pulse rates.


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Textbook Question

Online Gambling Some states now allow online gambling. As a marketing manager for a casino, you need to determine the percentage of adults in those states who gamble online. How many adults must you survey in order to be 99% confident that your estimate is in error by no more than two percentage points?


b. Assume that 18% of all adults gamble online (based on 2017 data from a Gambling Commission study in Great Britain).

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Textbook Question

7. FRESHMAN 15 Here is a sample of amounts of weight change (kg) of college students in their freshman year (from Data Set 13 “Freshman 15” in Appendix B): 11, 3, 0, –2, where –2 represents a loss of 2 kg and positive values represent weight gained. Here are ten bootstrap samples:

{11, 11, 11, 0}, {11, –2, 0, 11}, {11, –2, 3, 0}, {3, –2, 0, 11}, {0, 0, 0, 3}, {3, –2, 3, –2}, {11, 3, –2, 0}, {–2, 3, –2, 3}, {–2, 0, –2, 3}, {3, 11, 11, 11}.

b. Using only the ten given bootstrap samples, construct an 80% confidence interval estimate of the standard deviation of the weight changes for the population.

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Textbook Question

Comparing Waiting Lines


The values listed below are waiting times (in minutes) of customers at the Bank of Providence, where customers may enter any one of three different lines that have formed at three teller windows. Construct a 95% confidence interval for the population standard deviation sigma.

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Textbook Question

Critical Thinking. In Exercises 17–28, use the data and confidence level to construct a confidence interval estimate of p, then address the given question.


OxyContin The drug OxyContin (oxycodone) is used to treat pain, but it is dangerous because it is addictive and can be lethal. In clinical trials, 227 subjects were treated with OxyContin and 52 of them developed nausea (based on data from Purdue Pharma L.P.).


b. Compare the result from part (a) to this 95% confidence interval for 5 subjects who developed nausea among the 45 subjects given a placebo instead of OxyContin: . What do you conclude?

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