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Chapter 8: Problem 65

Find a \(90 \%\) one-sided upper confidence bound for the population mean \(\mu\) for these values: a. \(n=40, s^{2}=65, \bar{x}=75\) b. \(n=100, s=2.3, \bar{x}=1.6\)

Short Answer

Expert verified
b. For a sample with n=100, sample standard deviation s=2.3, and sample mean x虆=1.6?

Step by step solution

01

a. Finding the t-score for 90% confidence level and \(n=40\)

To find the t-score in case a, look up the one-sided t-distribution table for a 90% confidence level and 39 degrees of freedom (since \(n-1=40-1=39\)). The t-score is approximately 1.685.
02

b. Finding the t-score for 90% confidence level and \(n=100\)

To find the t-score in case b, look up the one-sided t-distribution table for a 90% confidence level and 99 degrees of freedom (since \(n-1=100-1=99\)). The t-score is approximately 1.660. Next, we will use these t-scores to calculate the margin of error and find the one-sided upper confidence bound for both cases.
03

a. Calculate the margin of error and upper confidence bound for \(\mu\) using \(n=40, s^{2}=65, \bar{x}=75\)

Given the values of \(n\), \(s^{2}\), and \(\bar{x}\), we can plug them and the t-score calculated in the previous step into the formula as follows: Margin of Error = \(t \times \frac{s}{\sqrt{n}} = 1.685 \times \frac{\sqrt{65}}{\sqrt{40}} = 1.685 \times \frac{\sqrt{65}}{6.32} \approx 2.93\) The 90% one-sided upper confidence bound for the population mean, 渭, is: \(\bar{x} - \text{Margin of Error} = 75 - 2.93 = 72.07\)
04

b. Calculate the margin of error and upper confidence bound for \(\mu\) using \(n=100, s=2.3, \bar{x}=1.6\)

Given the values of \(n\), \(s\), and \(\bar{x}\), we can plug them and the t-score calculated in the previous step into the formula as follows: Margin of Error = \(t \times \frac{s}{\sqrt{n}} = 1.660 \times \frac{2.3}{\sqrt{100}} = 1.660 \times \frac{2.3}{10} \approx 0.382\) The 90% one-sided upper confidence bound for the population mean, 渭, is: \(\bar{x} - \text{Margin of Error} = 1.6 - 0.382 = 1.218\) Finally, we have the 90% one-sided upper confidence bounds for the population means \(\mu\): a. \(72.07\) b. \(1.218\)

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Most popular questions from this chapter

College graduates are getting more for their degrees as starting salaries rise. To compare the starting salaries of college graduates majoring in chemical engineering and computer science, random samples of 50 recent college graduates in each major were selected and the following information obtained. $$\begin{array}{lll}\text { Major } & \text { Mean } & \text { SD } \\\\\hline \text { Chemical engineering } & \$ 53,659 & 2225 \\\\\text { Computer science } & 51,042 & 2375\end{array}$$ a. Find a point estimate for the difference in starting salaries of college students majoring in chemical engineering and computer science. What is the margin of error for your estimate? b. Based upon the results in part a, do you think that there is a significant difference in starting salaries for chemical engineers and computer scientists? Explain.

Independent random samples of \(n_{1}=50\) and \(n_{2}=60\) observations were selected from populations 1 and \(2,\) respectively. The sample sizes and computed sample statistics are given in the table: $$\begin{array}{lcc} & \multicolumn{2}{c} {\text { Population }} \\\\\cline { 2 - 3 } & 1 & 2 \\\\\hline \text { Sample Size } & 5 & 60 \\\\\text { Sample Mean } & 100.4 & 96.2 \\\\\text { Sample Standard Deviation } & 0.8 & 1.3\end{array}$$ Find a \(90 \%\) confidence interval for the difference in population means and interpret the interval.

In an article in the Annals of Botany, a researcher reported the basal stem diameters of two groups of dicot sunflowers: those that were left to sway freely in the wind and those that were artificially supported. \({ }^{18}\) A similar experiment was conducted for monocot maize plants. Although the authors measured other variables in a more complicated experimental design, assume that each group consisted of 64 plants (a total of 128 sunflower and 128 maize plants). The values shown in the table are the sample means plus or minus the standard error. $$\begin{array}{l|c|c} & \text { Sunflower } & \text { Maize } \\\\\hline \text { Free-Standing } & 35.3 \pm .72 & 16.2 \pm .41 \\\\\text {Supported } & 32.1 \pm .72 & 14.6 \pm .40\end{array}$$ Use your knowledge of statistical estimation to compare the free-standing and supported basal diameters for the two plants. Write a paragraph describing your conclusions, making sure to include a measure of the accuracy of your inference.

Does Mars, Incorporated use the same proportion of red candies in its plain and peanut varieties? A random sample of 56 plain M\&M'S contained 12 red candies, and another random sample of 32 peanut M\&M'S contained 8 red candies. a. Construct a \(95 \%\) confidence interval for the difference in the proportions of red candies for the plain and peanut varieties. b. Based on the confidence interval in part a, can you conclude that there is a difference in the proportions of red candies for the plain and peanut varieties? Explain.

The first day of baseball comes in late March, ending in October with the World Series. Does fan support grow as the season goes on? Two CNN/USA Today/Gallup polls, one conducted in March and one in November, both involved random samples of 1001 adults aged 18 and older. In the March sample, \(45 \%\) of the adults claimed to be fans of professional baseball, while \(51 \%\) of the adults in the November sample claimed to be fans. \({ }^{13}\) a. Construct a \(99 \%\) confidence interval for the difference in the proportion of adults who claim to be fans in March versus November. b. Does the data indicate that the proportion of adults who claim to be fans increases in November, around the time of the World Series? Explain.
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