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Chapter 10: Problem 1

Briefly explain the meaning of independent and dependent samples. Give one example of each.

Short Answer

Expert verified
Independent samples are unrelated and chosen separately. For example, selecting students from two different grades to study. Dependent samples are related or matched, with one sample influencing the other. An example would be recording a student's grades before and after a tutoring program.

Step by step solution

01

Definition of Independent Samples

Independent samples are those where the sampling units for one sample do not affect which sampling units make up the other sample. The two samples are typically selected randomly from the same population, and the selection of individuals in one sample does not influence the selection in the other.
02

Example of Independent Samples

For example, in a school, one could choose one sample of students from 9th grade and another sample from 10th grade. The students in 9th grade are selected without any consideration of the students in 10th grade, making these two independent samples.
03

Definition of Dependent Samples

Dependent samples, on the other hand, are those where the value of one sample may influence or affect the value of the other sample. These are often paired or matched samples.
04

Example of Dependent Samples

For instance, consider a study where students' grades are recorded before and after a tutoring program. The same students are tested both times, making the samples (grades before and after) dependent.

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Most popular questions from this chapter

The manufacturer of a gasoline additive claims that the use of this additive increases gasoline mileage. A random sample of six cars was selected, and these cars were driven for 1 week without the gasoline additive and then for 1 week with the gasoline additive. The following table gives the miles per gallon for these cars without and with the gasoline additive. $$ \begin{array}{l|cccccc} \hline \text { Without } & 24.6 & 28.3 & 18.9 & 23.7 & 15.4 & 29.5 \\ \hline \text { With } & 26.3 & 31.7 & 18.2 & 25.3 & 18.3 & 30.9 \\ \hline \end{array} $$ a. Construct a \(99 \%\) confidence interval for the mean \(\mu_{d}\) of the population paired differences, where a paired difference is equal to the miles per gallon without the gasoline additive minus the miles per gallon with the gasoline additive. b. Using the \(2.5 \%\) significance level, can you conclude that the use of the gasoline additive increases the gasoline mileage?

An economist was interested in studying the impact of the recession on dining out, including drive-thru meals at fast food restaurants. A random sample of forty-eight families of four with discretionary incomes between \(\$ 300\) and \(\$ 400\) per week indicated that they reduced their spending on dining out by an average of \(\$ 31.47\) per week, with a sample standard deviation of \(\$ 10.95\). Another random sample of 42 families of five with discretionary incomes between \(\$ 300\) and \(\$ 400\) per week reduced their spending on dining out by an average \(\$ 35.28\) per week, with a sample standard deviation of \(\$ 12.37\). (Note that the two groups of families are differentiated by the number of family members.) Assume that the distributions of reductions in weekly dining-out spendings for the two groups have the same population standard deviation. a. Construct a \(90 \%\) confidence interval for the difference in the mean weekly reduction in diningout spending levels for the two populations. b. Using the \(5 \%\) significance level, can you conclude that the average weekly spending reduction for all families of four with discretionary incomes between \(\$ 300\) and \(\$ 400\) per week is less than the average weekly spending reduction for all families of five with discretionary incomes between \(\$ 300\) and \(\$ 400\) per week?

A sample of 500 observations taken from the first population gave \(x_{1}=305\). Another sample of 600 observations taken from the second population gave \(x_{2}=348\). a. Find the point estimate of \(p_{1}-p_{2}\). b. Make a \(97 \%\) confidence interval for \(p_{1}-p_{2}\). c. Show the rejection and nonrejection regions on the sampling distribution of \(\hat{p}_{1}-\hat{p}_{2}\) for \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1}>p_{2} .\) Use a significance level of \(2.5 \% .\) d. Find the value of the test statistic \(z\) for the test of part \(\mathrm{c}\). e. Will you reject the null hypothesis mentioned in part \(\mathrm{c}\) at a significance level of \(2.5 \%\) ?

The following information was obtained from two independent samples selected from two populations with unknown but equal standard deviations. $$ \begin{array}{lll} n_{1}=55 & \bar{x}_{1}=90.40 & s_{1}=11.60 \\ n_{2}=50 & \bar{x}_{2}=86.30 & s_{2}=10.25 \end{array} $$ a. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\).

As mentioned in Exercise \(10.26\), a town that recently started a single-stream recycling program provided 60-gallon recycling bins to 25 randomly selected households and 75-gallon recycling bins to 22 randomly selected households. The average total volumes of recycling over a 10 -week period were 382 and 415 gallons for the two groups, respectively, with standard deviations of \(52.5\) and \(43.8\) gallons, respectively. Suppose that the standard deviations for the two populations are not equal. a. Construct a \(98 \%\) confidence interval for the difference in the mean volumes of 10 -week recyclying for the households with the 60 - and 75 -gallon bins. b. Using the \(2 \%\) significance level, can you conclude that the average 10 -week recycling volume of all households having 60 -gallon containers is different from the average 10-week recycling volume of all households that have 75 -gallon containers? c. Suppose that the sample standard deviations were \(59.3\) and \(33.8\) gallons, respectively. Redo parts a and b. Discuss any changes in the results.
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