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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 the ones where the outcomes of one sample do not have any influence on the outcomes of the other, whereas, dependent samples are those where the outcomes of one sample do have a direct impact on the outcomes of another. Examples of independent samples could be studying the weight of men in a gym and women in a supermarket. Dependent samples could include studying the weight of people before and after a fitness program.

Step by step solution

01

Definition and Example of Independent Samples

Independent samples in statistics are samples that have no relationship between their occurrences. The outcomes or results of one do not affect the outcomes of the other. A typical example of independent sampling is measuring the weight of 30 randomly selected men in a gym and the weight of 30 other randomly selected women in a supermarket. These two samples are independent because the weights of men in the gym have no influence on the weights of women in the supermarket.
02

Definition and Example of Dependent Samples

Dependent samples, unlike independent ones, are where the outcomes of one sample do affect the outcomes of the other sample. This relationship defines the dependency between two sets. A common example of dependent sampling is measuring the weight of a group of individuals before they start a fitness program, and then measuring the weight of the same group of individuals after they have completed the program. Here, these two samples are dependent because the weight of the individuals after the program is dependent on their weight before the program.

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Most popular questions from this chapter

Conduct the following tests of hypotheses, assuming that the populations of paired differences are normally distributed. a. \(H_{0} \cdot \mu_{d f}=0, \quad H_{1}: \mu_{d} \neq 0, \quad n=26, \quad \bar{d}=9.6, \quad s_{d}=3.9, \quad \alpha=.05\) b. \(H_{0}: \mu_{d}=0, \quad H_{1}: \mu_{d}>0, \quad n=15, \quad \bar{d}=8.8, \quad s_{d}=4.7, \quad \alpha=.01\) c. \(H_{0}=\mu_{d}=0, \quad H_{1}: \mu_{d}<0, \quad n=20, \quad \bar{d}=-7.4, \quad s_{d}=2.3, \quad \alpha=.10\)

Maine Mountain Dairy claims that its 8-ounce low-fat yogurt cups contain, on average, fewer calories than the 8 -ounce low-fat yogurt cups produced by a competitor. A consumer agency wanted to check this claim. A sample of 27 such yogurt cups produced by this company showed that they contained an average of 141 calories per cup. A sample of 25 such yogurt cups produced by its competitor showed that they contained an average of 144 calories per cup. Assume that the two populations are normally distributed with population standard deviations of \(5.5\) and \(6.4\) calories, repectively. a. Make a \(98 \%\) confidence interval for the difference between the mean number of calories in the 8-ounce low-fat yogurt cups produced by the two companies. b. Test at a \(1 \%\) significance level whether Maine Mountain Dairy's claim is true. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=05\) ? What if \(\alpha=.025\) ?

As mentioned in Exercise \(10.31\), Quadro Corporation has two supermarkets in a city. The company's quality control department wanted to check if the customers are equally satisfied with the service provided at these two stores. A sample of 380 customers selected from Supermarket I produced a mean satisfaction index of \(7.6\) (on a scale of 1 to 10,1 being the lowest and 10 being the highest) with a standard deviation of \(.75 .\) Another sample of 370 customers selected from Supermarket II produced a mean satisfaction index of \(8.1\) with a standard deviation of \(.59 .\) Assume that the customer satisfaction index for each supermarket has an unknown and different population standard deviation. a. Construct a \(98 \%\) confidence interval for the difference between the mean satisfaction indexes for all customers for the two supermarkets. b. Test at a \(1 \%\) significance level whether the mean satisfaction indexes for all customers for the itwo supermarkets are different. c. Suppose that the sample standard deviations were \(.88\) and \(.39\), respectively. Redo parts a and \(\mathrm{b}\). Discuss any changes in the results.

Explain when would you use the paired-samples procedure to make confidence intervals and test hypotheses.

The management at New Century Bank claims that the mean waiting time for all customers at its branches is less than that at the Public Bank, which is its main competitor. A business consulting firm took a sample of 200 customers from the New Century Bank and found that they waited an average of \(4.5\) minutes before being served. Another sample of 300 customers taken from the Public Bank showed that these customers waited an average of \(4.75\) minutes before being served. Assume that the standard deviations for the two populations are \(1.2\) and \(1.5\) minutes, respectively. a. Make a \(97 \%\) confidence interval for the difference between the population means. b. Test at a 2.5\% significance level whether the claim of the management of the New Century Bank is truc. c. Calculate the \(p\) -value for the test of part b. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=01 ?\) What if \(\alpha=05\) ?
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