91Ó°ÊÓ

In parts of the eastern United States, whitetail deer are a major nuisance to farmers and homeowners, frequently damaging crops, gardens, and landscaping. A consumer organization arranges a test of two of the leading deer repellents \(\mathrm{A}\) and \(\mathrm{B}\) on the market. Fifty-six unfenced gardens in areas having high concentrations of deer are used for the test. Twenty-nine gardens are chosen at random to receive repellent \(\mathrm{A}\), and the other 27 receive repellent \(\mathrm{B} .\) For each of the 56 gardens, the time elapsed between application of the repellent and the appearance in the garden of the first deer is recorded. For repellent \(\mathrm{A}\), the mean time is 101 hours. For repellent \(\mathrm{B}\), the mean time is 92 hours. Assume that the two populations of elapsed times have normal distributions with population standard deviations of 15 and 10 hours, respectively. a. Let \(\mu_{1}\) and \(\mu_{2}\) be the population means of elapsed times for the two repellents, respectively. Find the point estimate of \(\mu_{1}-\mu_{2}\). b. Find a \(97 \%\) confidence interval for \(\mu_{1}-\mu_{2}\). c. Test at the \(2 \%\) significance level whether the mean elapsed times for repellents \(\mathrm{A}\) and \(\mathrm{B}\) are different. Use both approaches, the critical-value and \(p\) -value, to perform this test.

Step by step solution

01

Find the point estimate

The point estimate of \( \mu_{1} - \mu_{2} \) is simply the difference between the sample means. Here, we have the mean time for repellent A as 101 hours and for repellent B as 92 hours. \n So, \( \mu_{1} - \mu_{2} = 101 - 92 = 9 \) hours.

02

Calculate the standard error

To create the confidence interval, we need to know the standard error. The standard error \( SE \) can be determined using the formula \[ SE = \sqrt{\frac{{s_{1}^{2}}}{n_{1}} + \frac{{s_{2}^{2}}}{n_{2}}} \] where \( s_{1} \) and \( s_{2} \) are the standard deviations, and \( n_{1} \) and \( n_{2} \) are the sample sizes. Here, \( s_{1} = 15 \), \( s_{2} = 10 \), \( n_{1} = 29 \), and \( n_{2} = 27 \). Substitute these values to calculate the standard error.

03

Find the confidence interval

Once we have the standard error, we can compute the 97% confidence interval for \( \mu_{1} - \mu_{2} \) which is given by the range \( (\mu_{1} - \mu_{2}) \pm Z_{\alpha/2} * SE \) . The value of \( Z_{\alpha/2} \) for a 97% confidence level corresponds to an alpha of 0.03 (because 1 - 0.97 = 0.03) and it is approximately 2.17.

04

Perform hypothesis testing

To test if the mean elapsed time for repellents A and B are different, we use a two-tailed test because we are not concerned about the direction of the difference. The null hypothesis (\( H_{0} \)) is \( \mu_{1} - \mu_{2} = 0 \) and the alternative hypothesis (\( H_{1} \)) is \( \mu_{1} - \mu_{2} \neq 0 \). Set the significance level (\( \alpha \)) to be 2%, and then calculate the test statistic (Z), which is equal to \( \frac{\mu_{1} - \mu_{2} - 0}{SE} \). Then compare this test statistic with the critical value for a 2% significance level (approximately ±2.58). We can also find the p-value by looking up the test statistic in a Z-table and doubling the result, as this is a two-tailed test.

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Key Concepts

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

Confidence Interval

A confidence interval is a range of values that is used to estimate the true value of a population parameter, like the difference in mean elapsed times in this exercise.

In this problem, we calculate a 97% confidence interval for the difference in means between the two deer repellents.
The confidence interval uses the sample data to create an interval estimate that aims to capture the true difference in population means with a certain level of confidence.
For a confidence interval, you need:

• The point estimate - this is the observed difference in means.
• The standard error - measures how much the sample mean is expected to fluctuate due to random sampling variation.
• The value from a standard distribution (here, the Z-score for 97% confidence level).

Together, these components provide an interval around the point estimate which we are 97% confident contains the true difference in means.
This means if we were to repeat the sampling many times, around 97% of the calculated intervals would include the true population parameter.

Point Estimate

The point estimate is a single value that serves as the best guess for a population parameter. In the context of this exercise, it indicates the difference between the means of two samples.
Here, the point estimate is obtained by calculating the difference between the average times the deer appeared after applying each repellent.

• The mean time for repellent A is 101 hours.
• The mean time for repellent B is 92 hours.

Thus, the point estimate for the difference in mean times is\[\hat{d} = 101 - 92 = 9 \text{ hours}\]This value of 9 hours suggests that, on average, Repellent A lasts 9 hours longer than Repellent B before deer appear.

Standard Error

The standard error (SE) is a measure that describes the variability or dispersion of a sample statistic. In hypothesis testing, it is crucial for determining the confidence interval and performing tests of significance.
To calculate the SE for the difference in sample means, the following formula is used:\[SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]Where:

• \( s_1 = 15 \) hours, and \( s_2 = 10 \) hours are the standard deviations for repellents A and B, respectively.
• \( n_1 = 29 \), and \( n_2 = 27 \) are the sample sizes for repellents A and B.

The SE represents how much the point estimate (difference between sample means) might vary if the experiment were to be repeated multiple times. The smaller the SE, the more reliable the estimate.

Normal Distribution

The concept of normal distribution plays a significant role in hypothesis testing and the construction of confidence intervals.
Normal distribution is a probability distribution that is symmetric around the mean and is shaped like a bell curve. Due to its symmetry, it provides properties that allow for rigorous statistical analysis.

• In many cases, the distribution of sample means will follow a normal distribution, especially when the sample size is large (Central Limit Theorem).
• In this exercise, the problem explicitly states that the elapsed times follow a normal distribution.

Given this assumption, statistical tests using the normal distribution (such as Z-tests) can be used to infer about population parameters based on the sample data.
The normality assumption simplifies calculations and helps in interpreting the results of the testing, like determining where most data points lie within a certain number of standard deviations from the mean.