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A November 2011 Pew Research Center poll asked American social media users about their use of social media (such as Facebook, Twitter, MySpace, or LinkedIn). The study is based on a national telephone survey of 2277 adult social media users conducted from April 26 to May 22, 2011 (www.pewinternet. org/Reports/201 1/Why-Americans-Use-Social-Media/Main-reportaspx). According to this survey, \(16 \%\) of 30 - to 49 -year-old and \(18 \%\) of \(50-\) to 64 -year-old social media users cited connecting with others with common hobbies or interests as a major reason for using social networking sites. Suppose that this survey included 562 social media users in the 30 to 49 age group and 624 in the 50 to 64 age group. a. Let \(p_{1}\) and \(p_{2}\) be the proportions of all social media users in the age groups 30 to 49 years and 50 to 64 years, respectively, who will cite connecting with others with common hobbies or interests as a major reason for using social networking sites. Construct a \(95 \%\) confidence interval for \(p_{1}-p_{2}\). b. Using a \(1 \%\) significance level, can you conclude that \(p_{1}\) is different from \(p_{2}\) ? Use both the criticalvalue and the \(p\) -value approaches.

Step by step solution

01

Calculate Proportions

To calculate the proportion of each group, multiply the total number of users in each age group by the percentage that cited the reason. For 30-49 age group, \(p_{1} = 16\% * 562 / 100 = 89.92 \). For 50-64 age group, \(p_{2} = 18\% * 624 / 100 = 112.32 \).

02

Construct 95% Confidence Interval for the Difference of Proportions

The confidence interval is computed as \((p_{1}-p_{2}) \pm Z*sqrt{(p_{1}(1-p_{1}/n_{1}) + p_{2}(1-p_{2})/n_{2})} \), where \(Z = 1.96\) for a 95% confidence level, \(n_{1} = 562\) and \(n_{2} = 624\). Using the calculated values, the 95% confidence interval becomes \((89.92 - 112.32) \pm 1.96*sqrt{\((89.92*(1-89.92/562) + 112.32*(1-112.32/624))\}\)

03

Conduct Hypothesis Test

The null hypothesis \(H_{0}\) is that \(p_{1} = p_{2}\), and the alternative hypothesis \(H_{1}\) that \(p_{1} ≠ p_{2}\). We test the null hypothesis against the alternative using a 1% significance level. Both the critical value and \(p\)-value approaches are used. The test statistic \(z\) is calculated as \((p_{1} - p_{2}) / sqrt{P(1 - P)(1/n_{1} + 1/n_{2})}\), where \(P\) is the pooled proportion. If the calculated \(z\)-score is beyond the thresholds set by the significance level or if the \(p\)-value is less than the significance level, we reject the null hypothesis.

04

Interpret the Results

If the 95% confidence interval for the difference between proportions does not contain zero, it suggests that the proportions are indeed different. In the hypothesis test, if the null hypothesis is rejected, it implies that there is sufficient evidence at 1% level of significance to conclude that the proportions of social media users citing the reason are different in 30-49 and 50-64 age groups.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a crucial method for making inferences about population parameters based on sample data. It involves formulating a null hypothesis (\( H_{0} \)) stating there is no effect or difference, and an alternative hypothesis (\( H_{1} \)) suggesting there is an effect or difference. In our survey example, the null hypothesis is that the proportions of 30-49-year-olds and 50-64-year-olds using social media for connecting with others are equal: \( H_{0}: p_{1} = p_{2} \). The alternative hypothesis (\( H_{1} \)) states that these proportions are different: \( H_{1}: p_{1} eq p_{2} \). By calculating a test statistic and comparing it to a threshold defined by the significance level, we can determine whether to reject the null hypothesis.

Steps involve:

• Defining hypotheses clearly.
• Choosing a test statistic.
• Calculating the statistic using sample data.
• Comparing it against critical values or p-values to make a conclusion.

This systematic approach ensures statistical claims are made with evidence.

Significance Level

The significance level, denoted by \( \alpha \), is a threshold to determine when to reject the null hypothesis. It indicates the probability of rejecting the null hypothesis when it is true, known as a Type I error. In hypothesis testing, a common choice is \( 5\% \) or \( 1\% \), like in our survey case.

A \( 1\% \) significance level means you require strong evidence before concluding a difference exists, leading to more conservative results. The level chosen affects the critical value in hypothesis tests and influences the width of confidence intervals. A low significance level, such as \( 1\% \), ensures a high standard of evidence, reducing false positives but potentially increasing false negatives.

• Helps balance the risks of errors in decision-making.
• Influences the stringency of statistical conclusions.
• Determines the critical region in hypothesis testing.

Survey Data Analysis

Survey data analysis involves collecting, summarizing, and interpreting data from a survey to make meaningful conclusions. The survey in the exercise collects data via a national telephone survey to assess social media usage across different age groups.

Key steps in survey data analysis include:

• Data Collection: Ensuring it is representative and unbiased.
• Data Summarization: Using descriptive statistics like proportions.
• Inferential Statistics: Drawing conclusions about the population.

In the example, proportions of different age groups are analyzed to understand social media usage trends. Proper data analysis leads to more accurate and insightful conclusions, helping researchers to identify patterns and differences effectively.

Difference of Proportions

The difference of proportions is vital for comparing two distinct groups. In our survey, we compare the proportions of different age groups citing connecting for interests as a reason to use social media. This involves calculating the difference between the sample proportions \( p_{1} - p_{2} \).

To evaluate this difference, a confidence interval can be constructed. A \( 95\% \) confidence interval provides a range that likely contains the true difference between the population proportions.

• Calculated via standard error and critical value (e.g., \( Z = 1.96 \) for \( 95\% \) level).
• If the interval does not include zero, it suggests a real difference exists.

This method not only estimates the difference but also provides context for its statistical significance, aiding in robust decision-making in comparing proportions.