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Chapter 6: Problem 120

How Many Social Ties Do You Have? Most US adults have social ties with a large number of people, including friends, family, co-workers, and other acquaintances. It is nearly impossible for most people to reliably list all the people they know, but using a mathematical model, social analysts estimate that, on average, a US adult has social ties with 634 people. \(^{28}\) A survey of 1700 randomly selected US adults who are cell phone users finds that the average number of social ties for the cell phone users in the sample was 664 with a standard deviation of 778 . Does the sample provide evidence that the average number of social ties for a cell phone user is significantly different from 634 , the hypothesized number for all US adults? Define any parameters used and show all details of the test.

Short Answer

Expert verified
After calculating the value of the test statistic and comparing it with the critical value, the result will provide evidence either for or against the hypothesis that the average number of social ties for a cell phone user is significantly different from 634. The specifics of this answer will depend on the computed z-value.

Step by step solution

01

Parameters Definition

Define the parameters involved in the exercise. Here, the parameters are the population mean (\(渭\)), which is the average number of social ties for all US adults, estimated to be 634, and the sample mean (\(x虅\)), which is the average number of social ties for the sampled cell phone users, given as 664.
02

Hypothesis Formulation

Formulate the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)). Here, \(H_0: 渭 = 634\) (the average number of social ties for a cell phone user is the same as for all US adults) and \(H_1 : 渭 鈮� 634\) (the average number of social ties for a cell phone user is significantly different from all US adults).
03

Test Statistic Computation

Compute the test statistic using the formula \(z = \frac{x虅 - 渭}{蟽/ 鈭歯}\). Here, \(x虅 = 664\), \(渭 = 634\), \(蟽 = 778\) (standard deviation), and \(n = 1700\) (sample size). Plugging these values into the formula gives \(z = (664 - 634)/(778/鈭�1700)\). Calculate the value of z.
04

Critical Value Determination

Determine the critical value for a two-tailed test (since the alternative hypothesis is \(渭 鈮� 634\)) at a level of significance of 0.05. Looking up a standard normal table, the critical value \(Z_{0.025}\) is approximately 卤1.96.
05

Rejection Region Specification

Specify the rejection region. If the test statistic falls in the rejection region, then we reject the null hypothesis. The rejection region for a two-tailed test is \(z > Z_{0.025}\) or \(z < -Z_{0.025}\), i.e., \(z > 1.96\) or \(z < -1.96\).
06

Conclusion

Compare your computed test statistic with the critical value to decide whether to reject or fail to reject the null hypothesis. If \(z > 1.96\) or \(z < -1.96\), we reject \(H_0\) and conclude that the average number of social ties for a cell phone user is significantly different from 634, the hypothesized number for all US adults.

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Key Concepts

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

Understanding Population Mean
The concept of population mean is a fundamental aspect of statistics, representing the average of a set of values in a population. The population mean, often denoted by the Greek letter \( \mu \), is critical when making inferences about the overall population from sample data.
In the context of the provided exercise, the population mean is hypothesized to be 634, which is the average number of social ties for all US adults as estimated by social analysts. This number serves as a benchmark against which we compare our sample data to determine if there is a significant difference.
Understanding the population mean allows us to know the expected value around which individuals in a population should cluster. Knowing this can help in identifying significant deviations, hence providing insights into the applicability of surveyed data to the general population.
  • The population mean describes a whole group, as opposed to sample mean which focuses only on a portion of that group.
  • In hypothesis testing, \( \mu \) is essential for constructing our null and alternative hypotheses.
Role of Sample Standard Deviation
Sample standard deviation is a measure of the amount of variability or dispersion around the sample mean. It tells us how much the sample mean is expected to vary from the population mean, influenced by the data's spread. In this exercise, the sample standard deviation (noted as \( \sigma \)) was given as 778.
A larger standard deviation suggests more spread out data points, which can impact the precision of the test statistic we calculate. While the population mean helps us know where the data should be centered, the sample standard deviation provides critical insight into the consistency of the sample data.
  • Sample standard deviation gives us a sense of the reliability of the sample mean as an estimate of the population mean.
  • It's used in calculating the test statistic, a vital step in hypothesis testing.
  • A higher standard deviation indicates the data points are more spread out from the mean, impacting the confidence we have in sample conclusions.
Importance of Two-Tailed Test
The two-tailed test is an integral part of hypothesis testing. It is applied when we want to determine whether there are deviations in either direction (either significantly higher or lower) from the hypothesized population mean. In this scenario, the test aims to see if the average number of social ties for cell phone users is significantly different from 634.
This test divides the significance level between two tails of the normal distribution. By doing so, it assesses the possibility that the real mean could significantly deviate from the assumed mean in either direction.
  • A two-tailed test is used when the alternative hypothesis is non-directional, often denoted as \( H_1: \mu eq 634 \).
  • Properly applying a two-tailed test ensures that we are considering the full scope of potential deviations, both above and below the hypothesized population mean.
  • This type of test is more cautious and comprehensive, providing a robust mechanism for hypothesis validation.

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Most popular questions from this chapter

In Exercises 6.203 and \(6.204,\) use Stat Key or other technology to generate a bootstrap distribution of sample differences in means and find the standard error for that distribution. Compare the result to the standard error given by the Central Limit Theorem, using the sample standard deviations as estimates of the population standard deviations. Difference in mean commuting time (in minutes) between commuters in Atlanta and commuters in St. Louis, using \(n_{1}=500, \bar{x}_{1}=29.11,\) and \(s_{1}=20.72\) for Atlanta and \(n_{2}=500, \bar{x}_{2}=21.97,\) and \(s_{2}=14.23\) for St. Louis

Use the t-distribution and the given sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distributions are relatively normal. Test \(H_{0}: \mu_{A}=\mu_{B}\) vs \(H_{a}: \mu_{A} \neq \mu_{B}\) using the fact that Group A has 8 cases with a mean of 125 and a standard deviation of 18 while Group \(\mathrm{B}\) has 15 cases with a mean of 118 and a standard deviation of 14 .

The dataset ICUAdmissions, introduced in Data 2.3 on page \(69,\) includes information on 200 patients admitted to an Intensive Care Unit. One of the variables, Status, indicates whether each patient lived (indicated with a 0 ) or died (indicated with a 1 ). Use technology and the dataset to construct and interpret a \(95 \%\) confidence interval for the proportion of ICU patients who live.

Use a t-distribution and the given matched pair sample results to complete the test of the given hypotheses. Assume the results come from random samples, and if the sample sizes are small, assume the underlying distribution of the differences is relatively normal. Assume that differences are computed using \(d=x_{1}-x_{2}\). Test \(H_{0}: \mu_{1}=\mu_{2}\) vs \(H_{a}: \mu_{1}>\mu_{2}\) using the paired data in the following table: $$ \begin{array}{lllllllllll} \hline \text { Situation } & 1 & 125 & 156 & 132 & 175 & 153 & 148 & 180 & 135 & 168 & 157 \\ \text { Situation } & 2 & 120 & 145 & 142 & 150 & 160 & 148 & 160 & 142 & 162 & 150 \\ \hline \end{array} $$

In Exercises 6.109 to 6.111 , we examine the effect of different inputs on determining the sample size needed. Find the sample size needed to give, with \(95 \%\) confidence, a margin of error within 卤10 . Within 卤5 . Within 卤1 . Assume that we use \(\tilde{\sigma}=30\) as our estimate of the standard deviation in each case. Comment on the relationship between the sample size and the margin of error.
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