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Chapter 10: Problem 14

Use the following information to answer the next 15 exercises: Indicate if the hypothesis test is for a. independent group means, population standard deviations, and/or variances known b. independent group means, population standard deviations, and/or variances unknown c. matched or paired samples d. single mean e. two proportions f. single proportion A new WiFi range booster is being offered to consumers. A researcher tests the native range of 12 different routers under the same conditions. The ranges are recorded. Then the researcher uses the new WiFi range booster and records the new ranges. Does the new WiFi range booster do a better job?

Short Answer

Expert verified
c. matched or paired samples

Step by step solution

01

Identify the Hypothesis Type

In this problem, a researcher tests the native range of 12 routers and then tests the new range with a WiFi booster for the same 12 routers. Since the samples are taken from the same subjects (routers) before and after the intervention (WiFi booster), this indicates a situation where we are dealing with matched or paired samples.
02

Understand Matched or Paired Samples

Matched or paired samples occur when observations are not independent but are instead related in some way. In this example, each router's range is measured under two conditions: without the booster and with the booster.
03

Determine the Appropriate Test

Given that we have paired samples, the appropriate hypothesis test involves doing a paired t-test (also called a dependent t-test). This tests whether the mean difference between paired observations is zero or not, which relates directly to assessing the effectiveness of the WiFi range booster.

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Key Concepts

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

Understanding Hypothesis Tests
In statistics, a hypothesis test is a method used to determine whether there is enough evidence to reject a null hypothesis. Essentially, it's a way to test if a result from an experiment is statistically significant or if it occurred merely by chance. In our context, the researcher is testing whether the new WiFi range booster effectively increases the range of routers as opposed to there being no real difference. Hypothesis testing generally involves two hypotheses: the null hypothesis (\( H_0 \)) and the alternative hypothesis (\( H_a \)).
  • The null hypothesis often suggests that there is no effect or difference. For instance, "The WiFi booster does not change the range."
  • The alternative hypothesis suggests that there is an effect or difference, such as "The WiFi booster improves the range."
The goal of hypothesis testing is to determine which of these hypotheses is more plausible given the data collected from the experiment.
Explaining the Paired t-test
When dealing with paired samples, such as measuring the range of routers before and after using a WiFi booster, we use a paired t-test to analyze the data. This test helps assess whether the average difference between the paired observations is significantly different from zero. Here鈥檚 a simplified breakdown of the paired t-test:
  • Calculate the difference for each pair of observations. In this case, subtract the range measurement without the booster from the range with the booster for each router.
  • Compute the mean of these differences, which tells us the average change in range provided by the booster.
  • Determine the standard deviation of these differences to understand the variability.
  • Use these figures to calculate the t-statistic, which determines how far our sample differs from the null hypothesis of no change.
The t-statistic is then compared against a critical value from the t-distribution to decide if the results are statistically significant.
Understanding Mean Difference in Paired Samples
The mean difference is a central component in hypothesis tests involving paired samples. It is the average of the differences between paired observations. For our example with routers, it represents the average improvement in range when using the WiFi booster. When computing the mean difference, follow these easy steps:
  • For each pair, find the difference between the two conditions - with and without the booster.
  • Add all these differences together.
  • Divide this sum by the number of observations (i.e., pairs) to get the mean difference.
The mean difference tells us the typical boost in range the WiFi extender provides. In hypothesis testing, a significant mean difference suggests that the WiFi booster is effective in improving the range. If the mean difference is close to zero or not significantly different from zero after statistical analysis, it suggests that the booster does not have a significant effect.

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Most popular questions from this chapter

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. At a pre-conceived \(\alpha=0.05,\) what is your: a. Decision: b. Reason for the decision: c. Conclusion (write out in a complete sentence):

Use the following information to answer the next five exercises. A study was conducted to test the effectiveness of a software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in Table 10.21. The 鈥渂efore鈥� value is matched to an 鈥渁fter鈥� value, and the differences are calculated. The differences have a normal distribution. Test at the 1% significance level. $$ \begin{array}{|l|l|l|l|l|l|l|}\hline \text { Installation } & {\mathbf{A}} & {\mathbf{B}} & {\mathbf{C}} & {\mathbf{D}} & {\mathbf{E}} & {\mathbf{F}} & {\mathbf{G}} & {\mathbf{H}} \\ \hline \text { Before } & {3} & {6} & {4} & {2} & {5} & {8} & {2} & {6} \\ \hline \text { After } & {1} & {5} & {2} & {0} & {1} & {0} & {2} & {2} \\ \hline\end{array} $$ What is the p-value?

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. Sketch a graph of the situation. Label the horizontal axis. Mark the hypothesized difference and the sample difference. Shade the area corresponding to the p-value.

Use the following information to answer the next five exercises. A researcher is testing the effects of plant food on plant growth. Nine plants have been given the plant food. Another nine plants have not been given the plant food. The heights of the plants are recorded after eight weeks. The populations have normal distributions. The following table is the result. The researcher thinks the food makes the plants grow taller. $$\begin{array}{|l|l|l|}\hline \text { Plant Group } & {\text { Sample Mean Height of Plants (inches) }} & {\text { Population Standard Deviation }} \\\ \hline \text { Food } & {16} & {2.5} \\ \hline \text { No food } & {14} & {1.5} \\ \hline\end{array}$$ Is the population standard deviation known or unknown?

Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life expectancy was 47.6 years for whites born in 1900 and 33.0 years for nonwhites. Suppose that you randomly survey death records for people born in 1900 in a certain county. Of the 124 whites, the mean life span was 45.3 years with a standard deviation of 12.7 years. Of the 82 nonwhites, the mean life span was 34.1 years with a standard deviation of 15.6 years. Conduct a hypothesis test to see if the mean life spans in the county were the same for whites and nonwhites. In symbols, what is the random variable of interest for this test?
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