91Ó°ÊÓ

Eye Grease. Athletes performing in bright sunlight often smear black eye grease under their eyes to reduce glare. Does eye grease work? In one study, 16 student subjects took a test of visual sensitivity to light-and-dark contrast after three hours facing into bright sun, both with and without eye grease. This is a matched pairs design. If eye grease is effective, subjects will be more sensitive to contrast when they use eye grease. Here are the differences in sensitivity, with eye grease minus without eye grease:13 EYEGRS \begin{tabular}{|c|c|c|c|c|c|c|c|} \hline \(0.07\) & \(0.64\) & \(-0.12\) & \(-0.05\) & \(-0.18\) & \(0.14\) & \(-0.16\) & \(0.03\) \\ \hline \(0.05\) & \(0.02\) & \(0.43\) & \(0.24\) & \(-0.11\) & \(0.29\) & \(0.05\) & \(0.29\) \\\ \hline \end{tabular} We want to know whether eye grease increases sensitivity on the average. a. What are the null and alternative hypotheses? Say in words what mean \(\mu\) your hypotheses concern. b. Suppose that the subjects are an SRS of all young people with normal vision, that contrast differences follow a Normal distribution in this population, and that the standard deviation of differences is \(\sigma=0.22\). Carry out a test of significance.

Step by step solution

01

Define Hypotheses

First, define the null and alternative hypotheses in the context of this problem. The null hypothesis (H_0) is that the mean difference in sensitivity with and without eye grease is zero, indicating no effect of the eye grease. The alternative hypothesis (H_a) is that the mean difference in sensitivity with eye grease is greater than without it, indicating increased sensitivity.\[H_0: \mu = 0\]\[H_a: \mu > 0\]where \(\mu\) is the mean difference in sensitivity (eye grease - no eye grease).

02

Calculate the Mean Difference

Calculate the mean of the differences provided in the problem. These are the differences between sensitivity with eye grease and without eye grease:\[m = \frac{1}{16} \sum_{i=1}^{16} x_i \]Substitute the data values to find the mean.

03

Mean Difference Calculation

Add all the differences listed in the table:\[0.07 + 0.64 - 0.12 - 0.05 - 0.18 + 0.14 - 0.16 + 0.03 + 0.05 + 0.02 + 0.43 + 0.24 - 0.11 + 0.29 + 0.05 + 0.29 = 1.57\]Then divide by the number of data points (16):\[m = \frac{1.57}{16} = 0.098125\]

04

Calculate the Test Statistic

Using the sample mean \(m\), the standard deviation \(\sigma = 0.22\), and sample size \(n = 16\), calculate the test statistic using the formula for the one-sample z-test for means:\[Z = \frac{m - \mu_0}{\frac{\sigma}{\sqrt{n}}}\]where \(\mu_0 = 0\) under the null hypothesis.\[Z = \frac{0.098125 - 0}{\frac{0.22}{\sqrt{16}}} = \frac{0.098125}{0.055} \approx 1.7832\]

05

Find the p-value

Using the z-table, find the p-value corresponding to the calculated z-score. The z-score of 1.7832 corresponds to a cumulative probability. We find the p-value, which is the probability to the right of 1.7832 under the standard normal distribution.

06

Make a Decision

Determine if the p-value is less than the significance level (typically \(\alpha = 0.05\)). If the p-value is less than \(\alpha\), reject the null hypothesis. With a z-score of 1.7832, the corresponding p-value is approximately 0.037. Since 0.037 < 0.05, reject the null hypothesis.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Matched Pairs Design

In statistics, the matched pairs design is often used to analyze experiments where each subject is measured twice. This approach is particularly beneficial as it controls for variability between subjects. In the context of the eye grease study, each individual is observed both with and without eye grease, making them their own control. This means that each participant's response with and without the eye grease is compared, which increases the power of the hypothesis test. The primary advantage of this design is that it prevents confounding variables, as the same subjects are used in both conditions. In other words, any differences in outcome are likely due to the treatment effect (eye grease), rather than differences between subjects. This helps produce more reliable and valid results compared to unpaired designs.

Null and Alternative Hypotheses

The null and alternative hypotheses are central to hypothesis testing. They provide clear formulas for what we are testing. - The **null hypothesis** (\(H_0\)) represents a statement of no effect. In the eye grease experiment, it states that there is no difference in visual sensitivity when using eye grease, i.e., the mean difference (\(\mu\)) is zero: \(H_0: \mu = 0\).- Conversely, the **alternative hypothesis** (\(H_a\)) describes what we suspect might be true instead. For the eye grease study, it predicts a positive effect, meaning eye grease improves sensitivity. Thus, the mean difference is greater than zero: \(H_a: \mu > 0\).These hypotheses act as the foundation for determining which statistical tests to apply and interpreting the results of those tests.

Test Statistic Calculation

Calculating the test statistic is an essential step in hypothesis testing. It determines how many standard deviations our sample mean is from the hypothesized population mean under the null hypothesis. To find the test statistic in the eye grease experiment, we first calculate the mean difference of the sample. Given the data:- The total difference sum is 1.57.- Dividing by the number of observations (16), the mean (\(m\)) becomes approximately 0.098. Next, using the known standard deviation ($$\sigma = 0.22\() and sample size (\)n = 16\(), we apply the z-test formula:- \)Z = \frac{m - \mu_0}{\frac{\sigma}{\sqrt{n}}}\(,where \)\mu_0 = 0$.The calculated z-score, approximately 1.783, indicates how many standard deviations our sample mean is from the hypothesized mean.

P-value Interpretation

The p-value is crucial in deciding whether to support or reject the null hypothesis. It quantifies how likely it is to observe the test statistic or something more extreme, assuming the null hypothesis is true. For the eye grease study, a p-value of around 0.037 was found for the computed z-score of 1.783. - When the p-value is **less than** the chosen significance level (\(\alpha = 0.05\)), we reject the null hypothesis. It suggests that such an extreme result is unlikely due to random chance alone.- Conversely, if the p-value had been **greater than** 0.05, we would not have had enough evidence to reject \(H_0\).In this case, the p-value indicates a significant result, implying that eye grease likely improves visual sensitivity under glare conditions. This helps researchers conclude that their alternative hypothesis is supported by the data.