/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 45 The data sets represent simple r... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 45

The data sets represent simple random samples from a population whose mean is \(100 .\) $$ \begin{array}{rrrrr} & {\text { Data Set I }} \\ \hline 106 & 122 & 91 & 127 & 88 \\ \hline 74 & 77 & 108 & & \end{array} $$ $$ \begin{array}{rrrrr} \quad{\text { Data Set II }} \\ \hline 106 & 122 & 91 & 127 & 88 \\ \hline 74 & 77 & 108 & 87 & 88 \\ \hline 111 & 86 & 113 & 115 & 97 \\ \hline 122 & 99 & 86 & 83 & 102 \end{array} $$ $$ \begin{array}{rrrrr} {\text { Data Set III }} \\ \hline 106 & 122 & 91 & 127 & 88 \\ \hline 74 & 77 & 108 & 87 & 88 \\ \hline 111 & 86 & 113 & 115 & 97 \\ \hline 122 & 99 & 86 & 83 & 102 \\ \hline 88 & 111 & 118 & 91 & 102 \\ \hline 80 & 86 & 106 & 91 & 116 \end{array} $$ (a) Compute the sample mean of each data set. (b) For each data set, construct a \(95 \%\) confidence interval about the population mean. (c) What effect does the sample size \(n\) have on the width of the interval? For parts \((d)-(e),\) suppose that the data value 106 was accidentally recorded as \(016 .\) (d) For each data set, construct a \(95 \%\) confidence interval about the population mean using the incorrectly entered data. (e) Which intervals, if any, still capture the population mean, 100? What concept does this illustrate?

Short Answer

Expert verified
Sample means and confidence intervals are computed for each data set. The effect of sample size on interval width and the robustness of intervals to incorrect data are illustrated.

Step by step solution

01

Compute the Sample Mean of Data Set I

Add all values in Data Set I together and divide by the number of values. \[ \text{Sample Mean} = \frac{106 + 122 + 91 + 127 + 88 + 74 + 77 + 108}{8} \]
02

Compute the Sample Mean of Data Set II

Add all values in Data Set II together and divide by the number of values. \[ \text{Sample Mean} = \frac{106 + 122 + 91 + 127 + 88 + 74 + 77 + 108 + 87 + 88 + 111 + 86 + 113 + 115 + 97 + 122 + 99 + 86 + 83 + 102}{20} \]
03

Compute the Sample Mean of Data Set III

Add all values in Data Set III together and divide by the number of values. \[ \text{Sample Mean} = \frac{106 + 122 + 91 + 127 + 88 + 74 + 77 + 108 + 87 + 88 + 111 + 86 + 113 + 115 + 97 + 122 + 99 + 86 + 83 + 102 + 88 + 111 + 118 + 91 + 102 + 80 + 86 + 106 + 91 + 116}{30} \]
04

Construct a 95% Confidence Interval for Data Set I

Calculate the standard deviation and use the formula for the confidence interval. \[ \text{Confidence Interval} = \text{Sample Mean} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]
05

Construct a 95% Confidence Interval for Data Set II

Calculate the standard deviation and use the formula for the confidence interval. \[ \text{Confidence Interval} = \text{Sample Mean} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]
06

Construct a 95% Confidence Interval for Data Set III

Calculate the standard deviation and use the formula for the confidence interval. \[ \text{Confidence Interval} = \text{Sample Mean} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}} \]
07

Effect of Sample Size on Width of Interval

Discuss how a larger sample size decreases the margin of error, leading to a narrower confidence interval.
08

Construct 95% Confidence Interval with Incorrect Data for Data Set I

Recompute the sample mean using the incorrect data and then calculate the confidence interval. Replace 106 with 16.
09

Construct 95% Confidence Interval with Incorrect Data for Data Set II

Recompute the sample mean using the incorrect data and then calculate the confidence interval. Replace 106 with 16.
10

Construct 95% Confidence Interval with Incorrect Data for Data Set III

Recompute the sample mean using the incorrect data and then calculate the confidence interval. Replace 106 with 16.
11

Determine Which Intervals Capture the Population Mean

Check the computed confidence intervals to see if they still include the true population mean of 100.
12

Explain the Concept Illustrated by These Intervals

Discuss which intervals still capture the population mean despite the error and explain the concept of robustness.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Mean Calculation
The sample mean is a crucial concept in statistics. It is the average of all the values in a data set and provides a good estimate of the population mean. To calculate it, you simply sum all the values and divide by the number of values. For example, in Data Set I, you would add 106, 122, 91, 127, 88, 74, 77, and 108, then divide by 8. This calculation is essential in creating confidence intervals and assessing overall data trends.
Effect of Sample Size
Sample size plays a significant role in statistical estimates. Larger sample sizes tend to provide more reliable and stable results. When you increase the sample size, you reduce the variability of the sample mean. This makes your confidence intervals narrower, giving you a more precise estimate of the population mean. In the given exercise, Data Set III, with the largest sample size, will have the narrowest confidence interval, showing the importance of having more data points.
Standard Deviation
Standard deviation measures the dispersion or spread of the data values from the mean. It's crucial for constructing confidence intervals. A smaller standard deviation indicates that the values are closely clustered around the mean, while a larger standard deviation shows more spread. The formula for standard deviation in a sample is \[ s = \sqrt{ \frac{ \sum (x_i - \overline{x})^2 }{n - 1} } \]. In the problem, calculating the standard deviation for each data set helps in assessing variability.
Margin of Error
The margin of error tells us how much the sample mean's estimate may deviate from the actual population mean. It depends on the sample size and standard deviation. The formula for the margin of error at a given confidence level is \[ E = t_{ \alpha/2} \cdot \frac{s}{ \sqrt{n} } \]. This means that a larger sample or a smaller standard deviation will reduce the margin of error, giving a tighter and more precise confidence interval.
Robustness of Statistical Estimates
Robustness in statistics refers to the degree to which an estimate remains unaffected by small deviations or errors in the data. For example, if a value in our data is mistakenly recorded wrong, a robust statistical estimate will still be close to the true population parameter. In this exercise, even when 106 is mistakenly recorded as 016, the effect on larger data sets (like Data Set III) may be minimal, illustrating the robustness of the confidence interval estimates with larger samples.

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Most popular questions from this chapter

Gallup polled 982 likely voters immediately preceding the 2014 North Carolina senate race. The results of the survey indicated that incumbent Kay Hagan had the support of \(47 \%\) of respondents, while challenger Thom Tillis had support of \(46 \% .\) The poll's margin of error was \(3 \% .\) Gallup suggested the race was too close to call. Use the concept of a confidence interval to explain what this means.

Construct the appropriate confidence interval. A simple random sample of size \(n=785\) adults was asked if they follow college football. Of the 785 surveyed, 275 responded that they did follow college football. Construct a \(95 \%\) confidence interval for the population proportion of adults who follow college football.

In a survey conducted by the marketing agency 11 mark, 241 of 1000 adults 19 years of age or older confessed to bringing and using their cell phone every trip to the bathroom (confessions included texting and answering phone calls). (a) What is the sample in this study? What is the population of interest? (b) What is the variable of interest in this study? Is it qualitative or quantitative? (c) Based on the results of this survey, obtain a point estimate for the proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (d) Explain why the point estimate found in part (c) is a statistic. Explain why it is a random variable. What is the source of variability in the random variable? (e) Construct and interpret a \(95 \%\) confidence interval for the population proportion of adults 19 years of age or older who bring their cell phone every trip to the bathroom. (f) What ensures that the results of this study are representative of all adults 19 years of age or older?

Suppose you have two populations: Population \(\mathrm{A}-\) All students at Illinois State University \((N=21,000)\) and Population \(\mathrm{B}-\) All residents of the city of Homer Glen, IL \((N=21,000)\). You want to estimate the mean age of each population using two separate samples each of size \(n=75\). If you construct a \(95 \%\) confidence interval for each population mean, will the margin of error for population A be larger, the same, or smaller than the margin of error for population \(\mathrm{B}\) ? Justify your reasoning.

Indicate whether a confidence interval for a proportion or mean should be constructed to estimate the variable of interest. Justify your response. A developmental mathematics instructor wishes to estimate the typical amount of time students dedicate to studying mathematics in a week. She asks a random sample of 50 students enrolled in developmental mathematics at her school to report the amount of time spent studying mathematics in the past week.
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