### Hypothesis Testing - Example ProblemYou wish to test the following claim ($ H_a $) at a significance level of $ \alpha = 0.10 $.\[H_0: \mu = 66.2 \\H_a: \mu \ne 66.2\]You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size $ n = 10 $ with mean $ M = 43.4 $ and a standard deviation of $ SD = 18.5 $.1. **Test Statistic Calculation:** What is the test statistic for this sample? (Report answer accurate to three decimal places.) \[ \text{test statistic} = \_\_\_\_\_ \]2. **P-Value Calculation:** What is the p-value for this sample? (Report answer accurate to four decimal places.) \[ \text{p-value} = \_\_\_\_\_ \]3. **P-Value Comparison:** The p-value is... - O less than (or equal to) $ \alpha $ - O greater than $ \alpha $4. **Decision Rule:** This test statistic leads to a decision to... - O reject the null - O accept the null - O fail to reject the null5. **Final Conclusion:** As such, the final conclusion is that... - O There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. - O There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. - O The sample data support the claim that the population mean is not equal to 66.2. - O There is not sufficient sample evidence to support the claim that the population mean is not equal to 66.2.Note: This problem involves conducting a two-tailed t-test given that the population standard deviation is unknown and the sample size is small ($ n < 30 $). The test statistic for the t-test is calculated using the sample mean, sample standard deviation, and sample size. The p-value obtained from the t-distribution helps in making a decision about the null hypothesis.

Answered: Image /qna-images/answer/f2d272e0-d966-44b4-984d-2e0d0c323ab7.jpg

You wish to test the following claim (H,) at a significance level of a = 0.10. H,:µ= 66.2 H:µ + 66.2 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 10 with mean M = 43.4 and a standard deviation of SD = 18.5. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is. O less than (or equal to) a O greater than a This test statistic leads to a decision to. O reject the null O accept the null O fail to reject the null As such, the final conclusion is that. O There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O The sample data support the claim that the population mean is not equal to 66.2. O There is not sufficient sample evidence to support the claim that the population mean is not equal to 66.2.

You wish to test the following claim (H,) at a significance level of a = 0.10. H,:µ= 66.2 H:µ + 66.2 You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size n = 10 with mean M = 43.4 and a standard deviation of SD = 18.5. What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic = What is the p-value for this sample? (Report answer accurate to four decimal places.) p-value = The p-value is. O less than (or equal to) a O greater than a This test statistic leads to a decision to. O reject the null O accept the null O fail to reject the null As such, the final conclusion is that. O There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2. O The sample data support the claim that the population mean is not equal to 66.2. O There is not sufficient sample evidence to support the claim that the population mean is not equal to 66.2.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

### Hypothesis Testing - Example Problem

You wish to test the following claim ($ H_a $) at a significance level of $ \alpha = 0.10 $.

\[
H_0: \mu = 66.2 \\
H_a: \mu \ne 66.2
\]

You believe the population is normally distributed, but you do not know the standard deviation. You obtain a sample of size $ n = 10 $ with mean $ M = 43.4 $ and a standard deviation of $ SD = 18.5 $.

1. **Test Statistic Calculation:**

What is the test statistic for this sample? (Report answer accurate to three decimal places.)

\[
\text{test statistic} = \_\_\_\_\_
\]

2. **P-Value Calculation:**

What is the p-value for this sample? (Report answer accurate to four decimal places.)

\[
\text{p-value} = \_\_\_\_\_
\]

3. **P-Value Comparison:**

The p-value is...

- O less than (or equal to) $ \alpha $
- O greater than $ \alpha $

4. **Decision Rule:**

This test statistic leads to a decision to...

- O reject the null
- O accept the null
- O fail to reject the null

5. **Final Conclusion:**

As such, the final conclusion is that...

- O There is sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2.
- O There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 66.2.
- O The sample data support the claim that the population mean is not equal to 66.2.
- O There is not sufficient sample evidence to support the claim that the population mean is not equal to 66.2.

Note: This problem involves conducting a two-tailed t-test given that the population standard deviation is unknown and the sample size is small ($ n < 30 $). The test statistic for the t-test is calculated using the sample mean, sample standard deviation, and sample size. The p-value obtained from the t-distribution helps in making a decision about the null hypothesis.

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