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Chapter 7: Problem 48

According to a 2018 Pew Research Center report on social media use, \(28 \%\) of American adults use Instagram. Suppose a sample of 150 American adults is randomly selected. We are interested in finding the probability that the proportion of the sample who use Instagram is greater than \(30 \%\). a. Without doing any calculations, determine whether this probability will be greater than \(50 \%\) or less than \(50 \%\). Explain your reasoning. b. Calculate the probability that the sample proportion is \(30 \%\) or more.

Short Answer

Expert verified
a. The probability that the proportion of the sample who use Instagram is greater than 30% is less than 50% because 30% is greater than the population proportion of 28%. b. The calculated probability for more than 30% using Instagram in the selected sample needs the final value from step 5.

Step by step solution

01

Reasoning

Given the population proportion (p) of American adults who use Instagram is 28%. If we are randomly selecting American adults, it is more likely that the sample proportion will be close to the population proportion rather than far from it. Therefore, the probability that more than 30% of the sampled adults using Instagram would be less than 50% since 30% is greater than the given population proportion of 28%.
02

Set Up The Variables

To solve for part b of the exercise, we need to set up the variables appropriate for this statistics problem. We know that the sample size (n) = 150, the population proportion (p) = 0.28 and we need to find the sample proportion greater than or equal to 0.30.
03

Estimate the Mean and Standard Deviation

First, calculate the mean (渭) and standard deviation (蟽) of the sample distribution. The mean of the distribution of sample proportions equals to the population proportion, 渭 = p = 0.28. The standard deviation of the distribution of sample proportions is calculated as \(\sqrt{ ( p * ( 1 - p ) ) / n }\) = \(\sqrt{ (0.28 * 0.72) / 150 }\).
04

Find the Z-Score

After obtaining the values of 渭 and 蟽, calculate the Z-score for the sample proportion (.30). The formula is Z = (sample proportion - 渭) / 蟽. By substituting the values we will find the Z score is equal to: (0.30 - 0.28) / 蟽.
05

Calculate the Probability

Finally, using the calculated Z score and the standard normal distribution table (Z-table) or a calculator with normal distribution probability capability, the probability can be found that the sample proportion is 30% or more. That is P(Z 鈮� calculated Z value).

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Most popular questions from this chapter

According to a 2017 survey conducted by Netflix, \(46 \%\) of couples have admitted to "cheating" on their significant other by streaming a TV show ahead of their partner. Suppose a random sample of 80 Netflix subscribers is selected. a. What percentage of the sample would we expect have "cheated" on their partner? b. Verify that the conditions for the Central Limit Theorem are met. c. What is the standard error for this sample proportion? d. Complete the sentence: We expect ____ \(\%\) of streaming couples to admit to Netflix "cheating," give or take _____ \(\% .\)

A random sample of likely voters showed that \(49 \%\) planned to support Measure \(X\). The margin of error is 3 percentage points with a \(95 \%\) confidence level. a. Using a carefully worded sentence, report the \(95 \%\) confidence interva for the percentage of voters who plan to support Measure \(\mathrm{X}\). b. Is there evidence that Measure \(\mathrm{X}\) will fail? c. Suppose the survey was taken on the streets of Miami and the measure was a Florida statewide measure. Explain how that would affect your conclusion.

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$\begin{array}{|l|l|l|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\\\\hline & 936 & 607 \\ \hline\end{array}$$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$\begin{array}{|l|c|c|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\\\\hline & 94 & 61 \\ \hline\end{array}$$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.

From Formula \(7.2\), an estimate for margin of error for a \(95 \%\) confidence interval is \(m=2 \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\) where \(\mathrm{n}\) is the required sample size and \(\hat{p}\) is the sample proportion. Since we do not know a value for \(\hat{p}\), we use a conservative estimate of \(0.50\) for \(\hat{p}\). Replace \(\hat{p}\) with \(0.50\) in the formula and simplify.

\(1,3,5,7\), and 9 are odd and 0,2 , 4,6, and 8 are even. Consider a 30 -digit line from a random number table. a. How many of the 30 digits would you expect to be odd on average? b. If you actually counted, would you get exactly the number you predicted in part a? Explain.
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