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Chapter 10: Problem 3

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 windshield treatment claims to repel water more effectively. Ten windshields are tested by simulating rain without the new treatment. The same windshields are then treated, and the experiment is run again. A hypothesis test is conducted.

Short Answer

Expert verified
The hypothesis test is for "matched or paired samples".

Step by step solution

01

Identify the Type of Samples

The problem description mentions that the same ten windshields are tested before and after being treated. This indicates that the samples are dependent since the same subjects (windshields) are used in both conditions.
02

Determine the Appropriate Hypothesis Test

Given that the samples are dependent and the same ten windshields are used for both tests, the hypothesis test falls under "matched or paired samples". This test is appropriate for experiments where the same subjects are tested under different conditions and we want to compare the means of two related groups.

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Key Concepts

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

Hypothesis Testing
Hypothesis testing is a statistical method used to make decisions based on data. When dealing with experiments or studies, it helps determine whether the observed effects are genuine or simply due to random chance.
The process involves comparing sample data against a hypothesis, like the claim from a windshield treatment company that their product repels water more effectively.
To test this claim, a hypothesis test includes:
  • Formulating a null hypothesis ( $$H_0$$ ) which represents the status quo or no effect, such as "The treatment has no effect on water repulsion."
  • Setting an alternative hypothesis ( $$H_a$$ ) indicating what you aim to prove, like "The treatment improves water repulsion."
  • Calculating a test statistic to evaluate the hypothesis, checking if the observed data deviate enough from what we'd expect if the null hypothesis was true.
  • Determining a significance level (commonly set at 0.05) to decide whether to reject or fail to reject the null hypothesis.
Hypothesis testing gives a structured framework to evaluate claims and make informed decisions based on statistical evidence.
Dependent Samples
Dependent samples, also known as matched or paired samples, occur when the same subjects are measured multiple times or under different conditions.
In the case of the windshield treatment, the same ten windshields are tested before and after applying the treatment. This setup ensures that each windshield serves as its own control, reducing variability unrelated to the treatment effect.
Characteristics of dependent samples include:
  • Each pair of observations is linked because they come from the same individual, subject, or entity before and after a treatment.
  • The data collection design helps identify changes directly attributable to different conditions since external factors are minimized by the matched design.
By using dependent samples in testing, one can gain more precise insights into how interventions affect the same subjects, leading to more accurate comparisons.
Mean Comparison
Mean comparison is a fundamental concept in statistics when evaluating differences between data sets or conditions.
In our windshield example, it involves comparing the average effectiveness of the water-repellent treatment before and after application.
For paired or dependent samples, this often involves:
  • Calculating the mean difference between paired observations, such as $$d = \bar{x}_{\text{treated}} - \bar{x}_{\text{untreated}}$$.
  • Performing a paired sample t-test or its non-parametric counterpart to analyze if the mean difference is statistically significant.
  • Understanding that the mean comparison seeks to identify a notable effect or change due to an applied condition or treatment.
The goal of such analyses is to see if any observed changes are likely due to the differing conditions rather than randomness, leading to meaningful conclusions about treatment effectiveness.

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Most popular questions from this chapter

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 It is believed that 70% of males pass their drivers test in the first attempt, while 65% of females pass the test in the first attempt. Of interest is whether the proportions are in fact equal.

Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 40 randomly chosen teenage girls and boys (20 of each) with cellular phones, the mean number of ring tones for the girls was 3.2 with a standard deviation of 1.5. The mean for the boys was 1.7 with a standard deviation of 0.8. Conduct a hypothesis test to determine if the means are approximately the same or if the girls鈥� mean is higher than the boys鈥� mean.

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 In a random sample of 100 forests in the United States, 56 were coniferous or contained conifers. In a random sample of 80 forests in Mexico, 40 were coniferous or contained conifers. Is the proportion of conifers in the United States statistically more than the proportion of conifers in Mexico?

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. $$\begin{array}{|l|l|l|}\hline \text { Plant Group } & {\text { Sample Mean Height of Plants (inches) }} & {\text { Population Standard Deviation }} \\\ \hline \text { Food } & {16} & {2.5} \\ \hline \text { No food } & {14} & {1.5} \\ \hline\end{array}$$ State the null and alternative hypotheses.

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. $$\begin{array}{|l|l|l|}\hline \text { Plant Group } & {\text { Sample Mean Height of Plants (inches) }} & {\text { Population Standard Deviation }} \\\ \hline \text { Food } & {16} & {2.5} \\ \hline \text { No food } & {14} & {1.5} \\ \hline\end{array}$$ Draw the graph of the p-value.
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