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Chapter 8: Problem 53

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.

Short Answer

Expert verified
Short Answer: The 95% confidence interval for the difference in proportions of red candies in plain and peanut M&M's is calculated using the sample proportions, the standard error, and a critical value (z = 1.96). By analyzing the confidence interval, we can determine if there is a significant difference in the proportions of red candies for the two varieties.

Step by step solution

01

Calculate sample proportions

Calculate the sample proportions for the plain M&M's (d鈧�) and the peanut M&M's (d鈧�) by dividing the number of red candies by the total number of candies in each sample: \(d鈧� = \frac{12}{56}\), \(d鈧� = \frac{8}{32}\)
02

Calculate difference in proportions

Calculate the difference of the sample proportions: \(\overset{\ \ }{p} = d鈧� - d鈧� = \frac{12}{56} - \frac{8}{32}\)
03

Calculate the standard error of the difference in proportions

Calculate the standard error of the difference in proportions using the formula: \(\overset{\ \ }{SE}(d鈧� - d鈧�) = \sqrt{\frac{d鈧�(1-d鈧�)}{n鈧亇 + \frac{d鈧�(1-d鈧�)}{n鈧倉}\) Plug in the values: \(\overset{\ \ }{SE}(\overset{\ \ }{p})= \sqrt{\frac{\left( \frac{12}{56}\right) \left( 1 - \frac{12}{56}\right)}{56} + \frac{\left( \frac{8}{32}\right) \left( 1 - \frac{8}{32}\right)}{32}}\)
04

Construct the 95% confidence interval

Use the standard error and a critical value (z = 1.96 for a 95% confidence interval) to construct the confidence interval: \(\overset{\ \ }{p} \pm z * \overset{\ \ }{SE}(\overset{\ \ }{p})\) Plug in the values: \(\left( \frac{12}{56} - \frac{8}{32} \right) \pm 1.96 * \sqrt{\frac{\left( \frac{12}{56}\right) \left( 1 - \frac{12}{56}\right)}{56} + \frac{\left( \frac{8}{32}\right) \left( 1 - \frac{8}{32}\right)}{32}}\)
05

Draw conclusions based on the confidence interval

Calculate the confidence interval and observe if it includes zero. If it does, we cannot conclude that there's a significant difference in the proportions of red candies for the plain and peanut varieties. If it doesn't include zero, we can conclude that there's a significant difference. Based on the calculated confidence interval, we can determine whether there is a difference in the proportions of red candies for the plain and peanut varieties.

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Most popular questions from this chapter

Of the 130 people in Exercise 8.37,65 were female and 65 were male. The means and standard deviation of their temperatures are shown below. $$\begin{array}{lrr} & \text { Men } & \text { Women } \\\\\hline \text { Sample Mean } & 98.11 & 98.39 \\\\\text { Standard Deviation } & 0.70 & 0.74\end{array}$$ Find a \(95 \%\) confidence interval for the difference in the average body temperatures for males versus females. Based on this interval, can you conclude that there is a difference in the average temperatures for males versus females? Explain.

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One of the major costs involved in planning a summer vacation is the cost of lodging. Even within a particular chain of hotels, costs can vary substantially depending on the type of room and the amenities offered. \(^{4}\) Suppose that we randomly select 50 billing statements from each of the computer databases of the Marriott, Radisson, and Wyndham hotel chains, and record the nightly room rates. $$\begin{array}{lccc} & \text { Marriott } & \text { Radisson } & \text { Wyndham } \\\\\hline \text { Sample average } & \$ 170 & \$ 145 & \$ 150 \\\\\text { Sample standard deviation } & 17.5 & 10 & 16.5\end{array}$$ a. Describe the sampled population(s). b. Find a point estimate for the average room rate for the Marriott hotel chain. Calculate the margin of error. c. Find a point estimate for the average room rate for the Radisson hotel chain. Calculate the margin of error. d. Find a point estimate for the average room rate for the Wyndham hotel chain. Calculate the margin of error. e. Display the results of parts \(\mathrm{b}, \mathrm{c},\) and d graphically, using the form shown in Figure \(8.5 .\) Use this display to compare the average room rates for the three hotel chains.

Refer to the Interpreting Confidence Intervals applet. a. Suppose that you have a random sample of size \(n=50\) from a population with unknown mean \(\mu\) and known standard deviation \(\sigma=35 .\) Calculate the half width of a \(95 \%\) confidence interval for \(\mu\). What would the width of this interval be? b. Use the button to create a single confidence interval for \(\mu\). What is the width of this interval? Compare your results to the calculation you did in part a.
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