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Chapter 4: Problem 61

Give null and alternative hypotheses for a population proportion, as well as sample results. Use StatKey or other technology to generate a randomization distribution and calculate a p-value. StatKey tip: Use "Test for a Single Proportion" and then "Edit Data" to enter the sample information. Hypotheses: \(H_{0}: p=0.7\) vs \(H_{a}: p<0.7\) Sample data: \(\hat{p}=125 / 200=0.625\) with \(n=200\)

Short Answer

Expert verified
Using the given null and alternative hypotheses and the sample data, a randomization distribution was generated via StatKey. The obtained p-value would then be used to either reject or fail to reject the null hypothesis. For instance, a p-value less than 0.05 would lead to rejecting \(H_{0}\), providing strong evidence for the alternative hypothesis that the population proportion is less than 0.7.

Step by step solution

01

Calculate Sample Proportion

Sample proportion (\(\hat{p}\)) is calculated by dividing the number of successful outcomes by the total number of trials. In this case, \(\hat{p} = 125 / 200 = 0.625\).
02

Using StatKey

To use StatKey, firstly select 'Proportion' under 'Randomization Test' since the given data deals with proportions. Then Edit Data and enter the hypothesized population proportion from \(H_{0} (0.7)\) and the results obtained from the sample data, in this case, successes = 125 , sample size = 200. Then click 'Calculate'.
03

Generate Randomization Distribution and Calculate p-value

After results are entered, StatKey will generate a simulation to create a randomization distribution. The p-value is calculated as the proportion of simulated results that are greater than or equal to the observed sample proportion.
04

Interpret the p-value

The p-value represents the chance of getting the sample data if the null hypothesis is true. If the p-value is low (say less than 0.05), the null hypothesis will be rejected, indicating that the sample data provides strong evidence that the actual population proportion is less than 0.7. If the p-value is high, the null hypothesis will not be rejected, meaning there is not enough evidence to suggest the actual population proportion is less than 0.7.

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Key Concepts

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

Population Proportion
The concept of 'population proportion' is critical in statistics as it represents the fraction of individuals in a population that have a particular attribute or characteristic. For instance, it could be the proportion of voters in a country who support a particular policy, or the proportion of consumers who prefer a specific brand of coffee.

To put it into context, if you have a town where 70 out of 100 households have solar panels, the population proportion of households with solar panels is 0.7. In statistical hypothesis testing, we often use population proportion when we're trying to draw conclusions about a population based on a sample from that population.
P-value
Understanding the 'p-value' can be a bit tricky, but it's essentially a measure of the strength of the evidence against the null hypothesis provided by the data. It tells us how likely it is to observe a sample statistic as extreme as the test statistic, under the assumption that the null hypothesis is true.

If the p-value is low, it suggests that such an extreme observed outcome is unlikely under the null hypothesis, pointing towards the alternative hypothesis. However, a high p-value indicates that the observed data is relatively normal under the null hypothesis, and thus, does not provide strong evidence against it. It's common to use a threshold (like 0.05) to decide if a p-value is 'low' or 'high'.
Randomization Distribution
The 'randomization distribution' plays a pivotal role in understanding how unusual our sample results are, within the context of the null hypothesis. It's created by simulating many samples, assuming the null hypothesis is true, and then seeing how the sample statistics (like sample proportions) are spread out.

This distribution helps us to place our actual observed sample statistic. If our observed statistic falls far into the tail of this distribution, it's a signal that our result is unusual 鈥� and perhaps the null hypothesis isn't the best explanation.
StatKey
When it comes to making statistical concepts practical, 'StatKey' is a fantastic tool. This is software designed specifically for teaching and learning statistics. It simulates the randomization test for a single proportion, among other tests. Students can enter their sample data, like the number of successes and the total number of trials, to see visual simulations of distributions and to calculate relevant statistics, such as the p-value, for making informed decisions on hypotheses.
Sample Proportion
Finally, let's talk about 'sample proportion'. This is similar to population proportion, but it deals with the sample drawn from the population rather than the entire population. It's the fraction of the sample that exhibits the characteristic we're interested in.

In our example, where 125 out of 200 people prefer a certain beverage, the sample proportion is 0.625. This is a crucial piece of the puzzle because it's the actual measurement we compare against what we expect from the population, to draw conclusions and test hypotheses.

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Most popular questions from this chapter

Female primates visibly display their fertile window, often with red or pink coloration. Do humans also do this? A study \(^{18}\) looked at whether human females are more likely to wear red or pink during their fertile window (days \(6-14\) of their cycle \()\). They collected data on 24 female undergraduates at the University of British Columbia, and asked each how many days it had been since her last period, and observed the color of her shirt. Of the 10 females in their fertile window, 4 were wearing red or pink shirts. Of the 14 females not in their fertile window, only 1 was wearing a red or pink shirt. (a) State the null and alternative hypotheses. (b) Calculate the relevant sample statistic, \(\hat{p}_{f}-\hat{p}_{n f}\), for the difference in proportion wearing a pink or red shirt between the fertile and not fertile groups. (c) For the 1000 statistics obtained from the simulated randomization samples, only 6 different values of the statistic \(\hat{p}_{f}-\hat{p}_{n f}\) are possible. Table 4.7 shows the number of times each difference occurred among the 1000 randomizations. Calculate the p-value.

Introductory statistics students fill out a survey on the first day of class. One of the questions asked is "How many hours of exercise do you typically get each week?" Responses for a sample of 50 students are introduced in Example 3.25 on page 244 and stored in the file ExerciseHours. The summary statistics are shown in the computer output below. The mean hours of exercise for the combined sample of 50 students is 10.6 hours per week and the standard deviation is 8.04 . We are interested in whether these sample data provide evidence that the mean number of hours of exercise per week is different between male and female statistics students. $$\begin{array}{lllrrrr} \text { Variable } & \text { Gender } & \text { N } & \text { Mean } & \text { StDev } & \text { Minimum } & \text{ Maximum } \\\\\text { Exercise } & \mathrm{F} & 30 & 9.40 & 7.41 & 0.00 & 34.00 \\\& \mathrm{M} & 20 & 12.40 & 8.80 & 2.00 & 30.00\end{array}$$ Discuss whether or not the methods described below would be appropriate ways to generate randomization samples that are consistent with \(H_{0}: \mu_{F}=\mu_{M}\) vs \(H_{a}: \mu_{F} \neq \mu_{M} .\) Explain your reasoning in each case. (a) Randomly label 30 of the actual exercise values with "F" for the female group and the remaining 20 exercise values with "M" for the males. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\). (b) Add 1.2 to every female exercise value to give a new mean of 10.6 and subtract 1.8 from each male exercise value to move their mean to 10.6 (and match the females). Sample 30 values (with replacement) from the shifted female values and 20 values (with replacement) from the shifted male values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) (c) Combine all 50 sample values into one set of data having a mean amount of 10.6 hours. Select 30 values (with replacement) to represent a sample of female exercise hours and 20 values (also with replacement) for a sample of male exercise values. Compute the difference in the sample means, \(\bar{x}_{F}-\bar{x}_{M}\) .

Interpreting a P-value In each case, indicate whether the statement is a proper interpretation of what a p-value measures. (a) The probability the null hypothesis \(H_{0}\) is true. (b) The probability that the alternative hypothesis \(H_{a}\) is true. (c) The probability of seeing data as extreme as the sample, when the null hypothesis \(H_{0}\) is true. (d) The probability of making a Type I error if the null hypothesis \(H_{0}\) is true. (e) The probability of making a Type II error if the alternative hypothesis \(H_{a}\) is true.

For each situation described, indicate whether it makes more sense to use a relatively large significance level (such as \(\alpha=0.10\) ) or a relatively small significance level (such as \(\alpha=0.01\) ). Using your statistics class as a sample to see if there is evidence of a difference between male and female students in how many hours are spent watching television per week.

Testing 100 right-handed participants on the reaction time of their left and right hands to determine if there is evidence for the claim that the right hand reacts faster than the left.
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