Refer To Example 14.9 In The Textbook. Complete The Following Hypothesis Test Given A Sample With N = 100 N = 100 N = 100 And X ˉ = 98.4 \bar{x} = 98.4 X ˉ = 98.4 . The Known Population Σ \sigma Σ Is 0.6.Null Hypothesis: H 0 : Μ = 98.6 H_0: \mu = 98.6 H 0 : Μ = 98.6 Alternate

Mar 10, 2025 by ADMIN 279 views

**Hypothesis Testing for a Known Population Standard Deviation**

Introduction

In statistics, hypothesis testing is a method used to determine whether there is enough evidence to reject a null hypothesis. In this article, we will refer to Example 14.9 in the textbook and complete a hypothesis test given a sample with $n = 100$ and $\bar{x} = 98.4$ . The known population standard deviation is 0.6. We will test the null hypothesis that the population mean is equal to 98.6 against an alternate hypothesis that the population mean is not equal to 98.6.

Null and Alternate Hypotheses

The null hypothesis is a statement of no effect or no difference, while the alternate hypothesis is a statement of an effect or a difference. In this case, the null hypothesis is:

H_0: μ = 98.6

This means that the population mean is equal to 98.6. The alternate hypothesis is:

H_a: μ ≠ 98.6

This means that the population mean is not equal to 98.6.

Given Information

We are given a sample with $n = 100$ and $\bar{x} = 98.4$ . The known population standard deviation is 0.6. We will use this information to complete the hypothesis test.

Test Statistic

The test statistic is a value that is used to determine whether the null hypothesis should be rejected. In this case, we will use the following formula to calculate the test statistic:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Calculating the Test Statistic

We will use the given information to calculate the test statistic. First, we need to calculate the sample standard deviation. Since we are given the population standard deviation, we can use the following formula to calculate the sample standard deviation:

s = σ / √n

where σ is the population standard deviation and n is the sample size.

s = 0.6 / √100 s = 0.6 / 10 s = 0.06

Now, we can calculate the test statistic:

t = (x̄ - μ) / (s / √n) t = (98.4 - 98.6) / (0.06 / √100) t = -0.2 / (0.06 / 10) t = -0.2 / 0.006 t = -33.33

Critical Region

The critical region is the range of values that would lead us to reject the null hypothesis. In this case, we are using a two-tailed test, which means that we will reject the null hypothesis if the test statistic is less than -tα/2 or greater than tα/2, where tα/2 is the critical value from the t-distribution with n-1 degrees of freedom.

Determining the Critical Value

We will use a t-distribution table to determine the critical value. Since we are using a two-tailed test, we will need to find the critical value for both the lower and upper tails. The degrees of freedom for this test is n-1 = 100-1 = 99.

Using a t-distribution table, we find that the critical value for a two-tailed test with 99 degrees of freedom and a significance level of 0.05 is:

tα/2 = 1.984

Making a Decision

We will now compare the test statistic to the critical value to make a decision about the null hypothesis.

t = -33.33 tα/2 = 1.984

Since the test statistic is less than the critical value, we reject the null hypothesis.

Conclusion

In this article, we completed a hypothesis test given a sample with $n = 100$ and $\bar{x} = 98.4$ . The known population standard deviation is 0.6. We tested the null hypothesis that the population mean is equal to 98.6 against an alternate hypothesis that the population mean is not equal to 98.6. We found that the test statistic was less than the critical value, so we rejected the null hypothesis.

Interpretation

The results of this hypothesis test indicate that the population mean is not equal to 98.6. This means that the sample mean of 98.4 is statistically significantly different from the population mean of 98.6.

Limitations

This hypothesis test has several limitations. First, the sample size is relatively small, which may affect the accuracy of the results. Second, the population standard deviation is known, which may not be the case in real-world applications. Finally, the null hypothesis is a specific value, which may not be the case in real-world applications.

Future Research

Future research could involve increasing the sample size to improve the accuracy of the results. Additionally, the population standard deviation could be estimated from the sample data, rather than being known. Finally, the null hypothesis could be a range of values, rather than a specific value.

References

[Textbook], Example 14.9.

Appendix

Calculations

The calculations for this hypothesis test are as follows:

t = (x̄ - μ) / (s / √n)
s = σ / √n
tα/2 = 1.984

Tables

The following table shows the critical values for the t-distribution with 99 degrees of freedom:

t	Probability
-1.984	0.025
1.984	0.025

Formulas

The following formulas were used in this hypothesis test:

t = (x̄ - μ) / (s / √n)
s = σ / √n
tα/2 = 1.984

Introduction

In the previous article, we completed a hypothesis test given a sample with $n = 100$ and $\bar{x} = 98.4$ . The known population standard deviation is 0.6. We tested the null hypothesis that the population mean is equal to 98.6 against an alternate hypothesis that the population mean is not equal to 98.6. In this article, we will answer some common questions about hypothesis testing.

Q: What is the purpose of hypothesis testing?

A: The purpose of hypothesis testing is to determine whether there is enough evidence to reject a null hypothesis. In other words, it is a method used to determine whether a sample of data is statistically significantly different from a known population.

Q: What is the difference between a null hypothesis and an alternate hypothesis?

A: The null hypothesis is a statement of no effect or no difference, while the alternate hypothesis is a statement of an effect or a difference. In the previous article, the null hypothesis was that the population mean is equal to 98.6, while the alternate hypothesis was that the population mean is not equal to 98.6.

Q: What is the significance level?

A: The significance level is the probability of rejecting the null hypothesis when it is true. It is usually denoted by the Greek letter alpha (α) and is set to a value such as 0.05.

Q: What is the critical region?

A: The critical region is the range of values that would lead us to reject the null hypothesis. In the previous article, we used a two-tailed test, which means that we rejected the null hypothesis if the test statistic was less than -tα/2 or greater than tα/2.

Q: How do I determine the critical value?

A: The critical value is determined using a t-distribution table or a statistical software package. The degrees of freedom for the test is n-1, where n is the sample size.

Q: What is the test statistic?

A: The test statistic is a value that is used to determine whether the null hypothesis should be rejected. It is calculated using the formula t = (x̄ - μ) / (s / √n), where x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the sample size.

Q: What is the difference between a one-tailed and a two-tailed test?

A: A one-tailed test is used when we are interested in determining whether the sample mean is greater than or less than a known population mean. A two-tailed test is used when we are interested in determining whether the sample mean is different from a known population mean.

Q: What is the p-value?

A: The p-value is the probability of observing a test statistic as extreme or more extreme than the one we obtained, assuming that the null hypothesis is true. It is a measure of the strength of the evidence against the null hypothesis.

Q: How do I interpret the results of a hypothesis test?

A: The results of a hypothesis test can be interpreted in several ways. If the null hypothesis is rejected, it means that the sample mean is statistically significantly different from the known population mean. If the null hypothesis is not rejected, it means that the sample mean is not statistically significantly different from the known population mean.

Q: What are some common mistakes to avoid when conducting a hypothesis test?

A: Some common mistakes to avoid when conducting a hypothesis test include:

Not checking the assumptions of the test
Not using the correct sample size
Not using the correct significance level
Not interpreting the results correctly