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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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