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Chapter 10: Problem 25

Assuming that the two populations have unequal and unknown population standard deviations, construct a \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) for the following. $$ \begin{array}{lll} n_{1}=48 & \bar{x}_{1}=.863 & s_{1}=.176 \\ n_{2}=46 & \bar{x}_{2}=.796 & s_{2}=.068 \end{array} $$

Short Answer

Expert verified
The \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) is \((0.008, 0.126)\).

Step by step solution

01

Calculate the degree of freedom

The dof is calculated using the Welch-Satterthwaite equation: \(dof = \frac{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^2} {(\frac{(s_1^2/n_1)^2}{n_1-1}+\frac{(s_2^2/n_2)^2}{n_2-1})}\). Substituting the given values, we get: \(dof = \frac{(\frac{0.176^2}{48} + \frac{0.068^2}{46})^2} {(\frac{(0.176^2/48)^2}{48-1}+\frac{(0.068^2/46)^2}{46-1})} = 81.78\) which rounds down to 81.
02

Obtain the t value

Using the t-distribution table, find the t-value for \(99\%\) confidence level (\(\alpha = 1 - 0.99 = 0.01\)) and \(dof = 81\). Since it's a two-tailed test, the \(\alpha\) value should be divided by 2, so look up for \(\alpha/2 = 0.005\) in the table. The corresponding t value is \(2.639\).
03

Calculate the confidence interval

The confidence interval is given by \(\bar{x}_1 - \bar{x}_2 \pm t_{\alpha/2, dof} \cdot \sqrt{\frac{s^2_1}{n_1} + \frac{s^2_2}{n_2}}\). Substituting the values: \(CI = 0.863 - 0.796 \pm 2.639 \cdot \sqrt{\frac{0.176^2}{48} + \frac{0.068^2}{46}} = 0.067 \pm 0.059\) So the \(99 \%\) confidence interval for \(\mu_{1}-\mu_{2}\) is \((0.008, 0.126)\).

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Most popular questions from this chapter

The following information was obtained from two independent samples selected from two populations with unknown but equal standard deviations. $$ \begin{array}{lll} n_{1}=55 & \bar{x}_{1}=90.40 & s_{1}=11.60 \\ n_{2}=50 & \bar{x}_{1}=86.30 & s_{2}=10.25 \end{array} $$ Test at a \(5 \%\) significance level if \(\mu_{1}\) is greater than \(\mu_{2}\).

Two local post offices are interested in knowing the average number of Christmas cards that are mailed out from the towns that they serve. A random sample of 80 households from Town A showed that they mailed an average of \(28.55\) Christmas cards with a standard deviation of \(10.30\). The corresponding values of the mean and standard deviation produced by a random sample of 58 households from Town \(\mathrm{B}\) were \(33.67\) and \(8.97\) Christmas cards. Assume that the distributions of the number of Christmas cards mailed by all households from both these towns have unknown and unequal population standard deviations. a. Construct a \(95 \%\) confidence interval for the difference in the average number of Christmas cards mailed by all households in these two towns. b. Using a \(10 \%\) significance level, can you conclude that the average number of Christmas cards mailed out by all households in Town A is different from the corresponding average for Town B?

The manager of a factory has devised a detailed plan for evacuating the building as quickly as possible in the event of a fire or other emergency. An industrial psychologist believes that workers actually leave the factory faster at closing time without following any system. The company holds fire drills periodically in which a bell sounds and workers leave the building according to the system. The evacuation time for each drill is recorded. For comparison, the psychologist also records the evacuation time when the bell sounds for closing time each day. A random sample of 36 fire drills showed a mean evacuation time of \(5.1\) minutes with a standard deviation of \(1.1\) minutes. A random sample of 37 days at closing time showed a mean evacuation time of \(4.2\) minutes with a standard deviation of \(1.0\) minute. a. Construct a \(99 \%\) confidence interval for the difference between the two population means. b. Test at a \(5 \%\) significance level whether the mean evacuation time is smaller at closing time than during fire drills. Assume that the evacuation times at closing time and during fire drills have equal but unknown population standard deviations.

A consumer organization tested two paper shredders, the Piranha and the Crocodile, designed for home use. Each of 10 randomly selected volunteers shredded 100 sheets of paper with the Piranha, and then another sample of 10 randomly selected volunteers each shredded 100 sheets with the Crocodile. The Piranha took an average of 203 seconds to shred 100 sheets with a standard deviation of 6 seconds. The Crocodile took an average of 187 seconds to shred 100 sheets with a standard deviation of 5 seconds. Assume that the shredding times for both machines are approximately normally distributed with equal but unknown standard deviations. a. Construct a \(99 \%\) confidence interval for the difference between the two population means. b. Using a \(1 \%\) significance level, can you conclude that the mean time taken by the Piranha to shred 100 sheets is higher than that for the Crocodile? c. What would your decision be in part \(\mathrm{b}\) if the probability of making a Type \(I\) error were zero? Explain.

Using data from the U.S. Census Bureau and other sources, www.nerdwallet.com estimated that considering only the households with credit card debts, the average credit card debt for U.S. households was \(\$ 15,523\) in 2014 and \(\$ 15,242\) in 2013 . Suppose that these estimates were based on random samples of 600 households with credit card debts in 2014 and 700 households with credit card debts in 2013 . Suppose that the sample standard deviations for these two samples were \(\$ 3870\) and \(\$ 3764\), respectively. Assume that the standard deviations for the two populations are unknown but equal. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the average credit card debts for all such households for the years 2014 and 2013 , respectively. What is the point estimate of \(\mu_{1}-\mu_{2}\) ? b. Construct a \(98 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Using a \(1 \%\) significance level, can you conclude that the average credit card debt for such households was higher in 2014 than in \(2013 ?\) Use both the \(p\) -value and the critical-value approaches to make this test.
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