A Sample Of 16 Items Provides A Sample Standard Deviation Of 9.5. Test The Following Hypotheses Using

A Sample Of 16 Items Provides A Sample Standard Deviation Of 9.5. Test The Following Hypotheses Using statistical methods to determine whether a particular claim about a population variance or standard deviation holds true. This scenario is common in quality control, research, and data analysis, where understanding the variability of data is crucial. In this article, we will explore the process of hypothesis testing involving sample standard deviations, focusing on how to set up hypotheses, perform calculations, and interpret results.

Understanding the Context and Purpose of Hypothesis Testing

Before diving into the calculations, it’s important to understand why hypothesis testing is employed in this context. When working with a sample standard deviation, the primary goal is often to infer whether the population standard deviation (or variance) matches a specified value or falls within certain bounds. For example, an engineer might want to verify if a manufacturing process maintains a consistent variability, or a researcher might test whether a new treatment affects the variability of outcomes.

Specifically, in our case, the sample contains 16 items, and the observed sample standard deviation is 9.5. The questions often posed are:

• Is the population variance equal to a specific value?


• Is it larger or smaller than the specified value?


• How confident can we be about the population variance given the sample data?

These questions are answered through hypothesis testing, which involves formulating null and alternative hypotheses, selecting an appropriate test statistic, and determining the statistical significance of the results.

Setting Up the Hypotheses

The first step in hypothesis testing is to clearly state the null hypothesis (H₀) and the alternative hypothesis (H₁ or Ha). These hypotheses depend on the question you are trying to answer.

Common Scenarios and Corresponding Hypotheses

• Testing if the population variance equals a specified value:

Suppose we want to test if the population variance σ² is equal to a hypothesized value σ₀².

• Null hypothesis (H₀): σ² = σ₀²
• Alternative hypothesis (H₁): σ² ≠ σ₀² (two-tailed)
• Testing if the population variance is greater than or less than a specified value:

For example, verifying if variability exceeds a certain threshold:

• H₀: σ² = σ₀²
• H₁: σ² > σ₀² (right-tailed) or σ² < σ₀² (left-tailed)

In our scenario, unless specified otherwise, we typically assume the goal is to test whether the population variance differs from a known or hypothesized variance.

Choosing the Appropriate Test: Chi-Square Test for Variance

Since the standard deviation and variance are related, and the sample size is small (n=16), the chi-square test is appropriate for testing hypotheses concerning the population variance.

Test Statistic

The test statistic for the chi-square test of variance is given by:

where:

• n = sample size


• s² = sample variance


• σ₀² = hypothesized population variance

This statistic follows a chi-square distribution with (n - 1) degrees of freedom if H₀ is true.

Calculating the Sample Variance

Given the sample standard deviation (s = 9.5), the sample variance is:

Performing the Hypothesis Test

Suppose we want to test whether the population variance is equal to a specific value, say σ₀² = 100. The process involves:

1. State hypotheses:

• H₀: σ² = 100


• H₁: σ² ≠ 100 (two-tailed)

1. Calculate the test statistic:

• n = 16


• s² = 90.25


• σ₀² = 100

Plug into the formula:

1. Determine the degrees of freedom:

• df = n - 1 = 15

1. Find the critical values:

Using chi-square distribution tables or software, identify the critical values at a chosen significance level (commonly α=0.05).

• For a two-tailed test at α=0.05 and df=15:


• Lower critical value (χ²₁): approximately 6.262


• Upper critical value (χ²₂): approximately 24.996

1. Decision rule:

• If χ² < χ²₁ or χ² > χ²₂, reject H₀.


• Otherwise, do not reject H₀.

1. Interpretation:

• Since 13.5375 is between 6.262 and 24.996, we fail to reject H₀ at the 5% significance level.

This indicates that, based on the sample data, there is not enough evidence to conclude that the population variance differs from 100.

Interpreting Results and Making Conclusions

The outcome of the hypothesis test hinges on the calculated test statistic and the critical values. If the test statistic falls into the rejection region, it suggests that the sample provides enough evidence to question the null hypothesis.

In our example:

• Failing to reject H₀ implies that the observed variability (standard deviation of 9.5) does not significantly differ from what is expected under the hypothesized variance of 100.


• Rejecting H₀ would indicate significant evidence that the population variance is different from 100, either larger or smaller depending on the test direction.

Factors Affecting the Test and Its Validity

Several factors influence the reliability of the chi-square test for variance:

• Sample size: Smaller samples have less power to detect differences, and the chi-square distribution is sensitive to sample size.
• Normality assumption: The chi-square test assumes the data are normally distributed. Deviations can affect the validity of the test.
• Significance level (α): The choice of α determines the threshold for rejecting H₀. Commonly, 0.05 is used, but stricter or more lenient levels can be chosen based on context.

Extensions and Related Tests

While the chi-square test is standard for variance, other related tests and extensions include:

1. One-sided Tests

Testing whether the variance is greater than or less than a certain value involves a one-tailed chi-square test, adjusting the critical region accordingly.

2. Confidence Intervals for Variance

Instead of hypothesis testing, you can construct confidence intervals for the population variance using chi-square distribution quantiles. This provides a range of plausible values for the true variance.

3. Testing for Equality of Variances

When comparing two or more populations, tests like Levene’s or Bartlett’s can assess whether variances are equal across groups.

Conclusion

Hypothesis testing involving the sample standard deviation provides a powerful method to infer properties about a population’s variability. In the scenario where a sample of 16 items yields a standard deviation of 9.5, the chi-square test offers a structured approach to assess whether the population variance aligns with a hypothesized value. Proper formulation of hypotheses, careful calculation of the test statistic, and interpretation based on the chi-square distribution facilitate informed decision-making in quality control, research, and data analysis.

Understanding these principles not only aids in making accurate inferences but also enhances the ability to critically evaluate data variability in various practical contexts. Whether verifying manufacturing consistency or analyzing experimental data, mastery of variance hypothesis testing is an essential skill for statisticians and data analysts alike.