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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 7.4.16b
Chapter 7, Problem 7.4.16b

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?


Verified step by step guidance
1
Step 1: Calculate the sample mean (x̄) of the given weights. Add all the weights together and divide by the total number of weights. Use the formula: x=∑xn, where n is the number of observations.
Step 2: Calculate the sample standard deviation (s). Use the formula: s=∑(x-x))2n-1, where x̄ is the sample mean and n is the number of observations.
Step 3: Determine the confidence interval using the formula for a t-distribution (since the sample size is small and the population standard deviation is unknown). The formula is: x±t⋅sn, where t is the critical value from the t-distribution table for the given confidence level and degrees of freedom (df = n - 1).
Step 4: Compare the calculated confidence interval to the one found in Exercise 14 in Section 7-3. Analyze whether the intervals overlap, are similar, or differ significantly.
Step 5: Interpret the comparison in the context of the problem. Discuss whether the weights of the pennies are consistent with the previous confidence interval and what this implies about the production process or variability in weights.

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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 with a specified level of confidence, typically 95% or 99%. It provides an estimate of uncertainty around a sample mean or proportion, allowing researchers to infer about the population from which the sample was drawn.
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Introduction to Confidence Intervals

Simple Random Sample

A simple random sample is a subset of individuals chosen from a larger population, where each individual has an equal chance of being selected. This method helps ensure that the sample is representative of the population, reducing bias and allowing for valid statistical inferences about the population based on the sample data.
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Sampling Distribution of Sample Proportion

Descriptive Statistics

Descriptive statistics summarize and describe the main features of a dataset, providing simple summaries about the sample and the measures. Common descriptive statistics include measures of central tendency (mean, median, mode) and measures of variability (range, variance, standard deviation), which help in understanding the distribution and characteristics of the data.
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Parameters vs. Statistics
Related Practice
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

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?


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Textbook Question

Smart Phone Apple is planning for the launch of a new and improved iPhone. The marketing team wants to know the worldwide percentage of consumers who intend to purchase the new model, so a survey is being planned. How many people must be surveyed in order to be 90% confident that the estimated percentage is within three percentage points of the true population percentage?


b. Assume that 11% of consumers have a smartphone and plan to upgrade to a new model.


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Textbook Question

Finite Population Correction Factor If a simple random sample of size n is selected without replacement from a finite population of size (n>0.05N), and the sample size is more than 5% of the population size , better results can be obtained by using the finite population correction factor, which involves multiplying the margin of error E by [Image]. Refer to the weights of the M&M candies in Data Set 38 “Candies” in Appendix B.


b. Use only the red M&Ms and treat that sample as a simple random sample selected from the population of the 345 M&Ms listed in the data set. Find the 95% confidence interval estimate of the mean weight of all 345 M&Ms. Compare the result to the actual mean of the population of all 345 M&Ms.


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Textbook Question

Caffeine in Soft Drinks Listed below are measured amounts of caffeine (mg per 12 oz of drink) obtained in one can from each of 20 brands (7UP, A&W Root Beer, Cherry Coke, . . . , TaB).


b. Given that Exercise 20 in Section 7-2 used the same data for a 99% confidence interval based on use of the t distribution, and given that the data do not appear to be from a normally distributed population, which confidence interval is likely to be better: The confidence interval from part (a) or the confidence interval found in Exercise 20 in Section 7-2?


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