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 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 of the first deer in the garden 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 approximately 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 a \(2 \%\) 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

Calculate point estimate of difference

The point estimate of \(\mu_{1}-\mu_{2}\) is given by the difference of sample means. Thus, \( \hat{\mu}_{1} - \hat{\mu}_{2} = 101 - 92 = 9 \) hours.

02

Calculate standard error

The standard error of the difference of sample means, given by \(\mathrm{SE} = \sqrt{\frac{s_{1}^{2}}{n_{1}} + \frac{s_{2}^{2}}{n_{2}}}\), where \(s_{1}\) and \(s_{2}\) are sample standard deviations and \(n_{1}\) and \(n_{2}\) are sample sizes. So, \(\mathrm{SE} = \sqrt{\frac{15^{2}}{29} + \frac{10^{2}}{27}} \approx 4.1\)

03

Compute the confidence interval

A \(97\%\) confidence interval for \(\mu_{1}-\mu_{2}\) is given by \((\hat{\mu}_{1}- \hat{\mu}_{2}) \pm z \cdot \mathrm{SE}\), where z is the critical value from the standard normal distribution corresponding to \(97\%\) confidence level (so, \(1.5\%\) in each tail of the curve). In this case, z-value is \(2.17\). The interval is \(9 \pm 2.17 \cdot 4.1\), which is \((0.07, 17.93)\) hours.

04

Set up the Hypothesis Test

Under both the critical value and \(p\) -value approach, we start by setting up the null hypothesis \(H_{0}: \mu_{1} = \mu_{2}\) and the alternative hypothesis \(H_{1}: \mu_{1} \neq \mu_{2}\)

05

Conduct Hypothesis Testing

We consider the standard normal z-distribution. The test statistic is \( z = (\hat{\mu}_{1} - \hat{\mu}_{2}) / \mathrm{SE} = 9 / 4.1 \approx 2.2\). Under the critical-value approach, we compare this test statistic to the critical z-value at the \(2\%\) significance level (1% in each tail), which is \(±2.58\). Since \(2.2\) does not fall in the rejection region, we fail to reject \(H_{0}\). Under the \(p\) -value approach, we calculate the \(p\)-value corresponding to obtained z-value from the standard normal distribution. Since \(p-value > \alpha = 2\%\), we fail to reject \(H_{0}\).

06

Interpret the results

We conclude that there is not sufficient evidence at the \(2\%\) level of significance to claim a difference in mean elapsed times between the two kinds of repellents. Therefore, it isn't clear that one repellent is better than the other.

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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 derived from sample data that is likely to contain the true population parameter with a certain level of confidence. In this exercise, we calculated a 97% confidence interval for the difference in means of elapsed times for the repellents, A and B. The expression used to calculate this interval is:\[ (\hat{\mu}_{1} - \hat{\mu}_{2}) \pm z \cdot \text{SE} \]where \(\hat{\mu}_{1} - \hat{\mu}_{2}\) represents the point estimate and SE is the standard error of the difference. The z-value corresponds to the chosen confidence level, which, for a 97% confidence interval, results in a z-value of about 2.17. Therefore, the interval was calculated as \(9 \pm 2.17 \cdot 4.1\), resulting in approximately \((0.07, 17.93)\) hours. This means we are 97% confident that the true difference in elapsed times between the two repellents falls within this range.

Point Estimate

The Point Estimate is a single value estimate of a population parameter. It provides an approximation of the true population value based on sample data. In this exercise, the point estimate of the difference in mean elapsed times (\(\mu_{1} - \mu_{2}\)) between the two repellents is calculated simply by taking the difference of the sample means:\[ \hat{\mu}_{1} - \hat{\mu}_{2} = 101 - 92 = 9 \text{ hours} \]This means that, on average, repellent A kept deer away for 9 hours longer than repellent B as per the samples taken. Remember that this is just an estimate—the actual population means may differ slightly.

Standard Error

The Standard Error (SE) measures the variability in the sampling distribution of a statistic, essentially providing an estimate of the standard deviation of that sampling distribution. In this scenario, SE is vital in calculating the confidence interval and conducting hypothesis tests. The formula used here is:\[ \text{SE} = \sqrt{\frac{s_{1}^2}{n_{1}} + \frac{s_{2}^2}{n_{2}}} \]where \(s_{1}\) and \(s_{2}\) are the standard deviations of the two groups, and \(n_{1}\) and \(n_{2}\) are their respective sample sizes. In the exercise, this turned out to be approximately 4.1 hours. A smaller SE implies more precision in estimating the population parameter.

Normal Distribution

Normal Distribution, often referred to as the bell curve, is a key concept in statistics and forms the basis for many statistical tests, including those in this exercise. It describes a symmetrical, bell-shaped distribution of data where most of the observations cluster around the central peak, and the probabilities for values taper off as they move away from the mean. In this problem, we assume that the elapsed times for deer repellents follow a normal distribution. This assumption allows us to use methods like confidence intervals and hypothesis testing on the means of these normal distributions. The normal distribution is central for calculating z-values, which help establish confidence intervals and test hypotheses. Here, the critical values for constructing the confidence interval and conducting the hypothesis test were derived from the standard normal distribution. Using the normal distribution assumptions leads to more accurate inference in many real-world applications, just as it does in this scenario.