91Ó°ÊÓ

As part of a study described in the report "I Can't Get My Work Done!" (harmon.ie/blog/i-cant-get-my-work \- done-how-collaboration-social-tools-drain-productivity, 2011, retrieved May 6,2017 ), people in a sample of 258 cell phone users ages 20 to 39 were asked if they use their cell phones to stay connected while they are in bed, and 168 said "yes." The same question was also asked of each person in a sample of 129 cell phone users ages 40 to \(49,\) and 61 said "yes." You might expect the proportion who stay connected while in bed to be higher for the 20 to 39 age group than for the 40 to 49 age group, but how much higher? a. Construct and interpret a \(90 \%\) large-sample confidence interval for the difference in the population proportions of cell phone users ages 20 to 39 and those ages 40 to 49 who say that they sleep with their cell phones. Interpret the confidence interval in context. Note that the sample sizes in the two groups - cell phone users ages 20 to 39 and those ages 40 to 49 -are large enough to satisfy the conditions for a large-sample test and a large-sample confidence interval for two population proportions. Even though the sample sizes are large enough, you can still use simulation-based methods. b. Use the output below to identify a \(90 \%\) bootstrap confidence interval for the difference in the population proportions of cell phone users ages 20 to 39 and those ages 40 to 49 who say that they sleep with their cell phones. c. Compare the confidence intervals you computed in Parts (a) and (b). Would your interpretation change using the bootstrap confidence interval compared with the large-sample confidence interval? Explain.

Step by step solution

01

Identify the proportions

We first need to identify the sample proportions for each age group. For the 20-39 age group, the proportion is: \(p_1 = \frac{168}{258}\) For the 40-49 age group, the proportion is: \(p_2 = \frac{61}{129}\)

02

Calculate the sample difference in proportions

Next, we find the difference between the sample proportions: \(\hat{p}_1 - \hat{p}_2 = \frac{168}{258} - \frac{61}{129}\)

03

Compute the large-sample confidence interval for the difference in proportions

Now, we can compute a 90% large-sample confidence interval for the difference in proportions using the formula: \(CI = (\hat{p}_1 - \hat{p}_2) \pm z \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}}\) Where \(z\) is the critical value for a 90% confidence interval, which is approximately 1.645. Plug in the values we previously calculated: \(CI = (\frac{168}{258} - \frac{61}{129}) \pm 1.645 \sqrt{\frac{\frac{168}{258} (1 - \frac{168}{258})}{258} + \frac{\frac{61}{129} (1 - \frac{61}{129})}{129}}\) Calculate the confidence interval. For part b, we would use the provided bootstrap output to determine another 90% confidence interval for the difference in proportions. This will involve looking at the results of the bootstrap simulation and determining the 5th and 95th percentiles for the difference.

04

Compare the confidence intervals and discuss the interpretation

Once we have both confidence intervals from part a and b, we can compare and evaluate if our interpretation based on the large-sample confidence interval would change if we used the bootstrap confidence interval. Consider the intervals' bounds, overlapping, and width when comparing them, and discuss whether the conclusions drawn from them are ultimately the same or different.

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Key Concepts

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

Difference in Proportions

In statistics, comparing two groups often involves looking at proportions. This means evaluating how a specific characteristic differs between these groups. For example, in our exercise, we want to understand if there's a difference in how people from two different age groups use their cell phones in bed.

To find the difference in proportions, we start by calculating the proportion for each group separately:

• The proportion for the 20-39 age group was calculated by dividing the number who said "yes" by the total number of respondents in that group, which was \(p_1 = \frac{168}{258}\).
• Similarly, for the 40-49 age group, the proportion \(p_2\) was \(\frac{61}{129}\).

The difference in proportions tells us whether one group tends to show a particular behavior more than the other. By subtracting these two proportions, we get a clearer picture of the disparity between the two age brackets.

Bootstrap Method

The Bootstrap method is a powerful statistical tool used to make inferences about a population from sample data. Unlike traditional methods requiring strict distribution assumptions, the bootstrap is more flexible. It relies on resampling with replacement from the original data.

Here's how it works:

• From the original sample, repeatedly draw new samples (with replacement).
• Calculate the statistic of interest, such as the difference in proportions, for each sample.
• Use the distribution of these statistics to estimate a confidence interval.

In our exercise, the bootstrap method provides an alternative to the large sample confidence interval for evaluating the difference in proportions. By using simulation-based outcomes, it gives insights that are often considered more robust, especially when dealing with odd-shaped data or smaller sample sizes than traditionally recommended.

Large Sample Test

The large sample test is a statistical approach primarily used when sample sizes are sufficiently large. This method assumes that the sampling distribution of the statistic under investigation will approximate a normal distribution due to the Central Limit Theorem.

In our exercise, we used a large sample test to compute a confidence interval for the difference in proportions between two age groups. Why a 90% confidence interval? This is a standard level selected to balance certainty and precision. The formula for creating this is:

• \[CI = (\hat{p}_1 - \hat{p}_2) \pm z \sqrt{\frac{\hat{p}_1 (1 - \hat{p}_1)}{n_1} + \frac{\hat{p}_2 (1 - \hat{p}_2)}{n_2}}\]
• The \(z\) value is a critical value from the standard normal distribution for a 90% confidence level, which is 1.645.

The advantage here is the simplicity and speed of calculation due to established statistical tables and the benefit of the Central Limit Theorem when sample sizes are deemed large enough.

Statistical Inference

Statistical inference involves making conclusions about a population based on information from a sample. It's a foundational concept in statistics allowing us to draw broader insights without surveying an entire population.

Two critical aspects of inference are estimation and hypothesis testing:

• Estimation: It involves determining the value of a population parameter (like a mean or proportion) from a sample statistic.
• Hypothesis testing: This determines whether enough evidence exists to support a particular claim about the population.

In our context, statistical inference uses both estimation (confidence intervals) and hypothesis testing to determine if there's a meaningful difference in phone usage between age groups. Statistical inference helps us make educated guesses (infer) about population behaviors and trends with some degree of confidence.