Please provide the following information. (a) What is the...

Please provide the following information. (a) What is the level of significance? State the null and alternate hypotheses. (b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution? (c) Find or estimate the $P$ -value of the sample test statistic. (d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence? (e) Interpret your conclusion in the context of the application. (f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application. Engineering: Jet Engines The fan blades on commercial jet engines must be replaced when wear on these parts indicates too much variability to pass inspection. If a single fan blade broke during operation, it could severely endanger a flight. A large engine contains thousands of fan blades, and safety regulations require that variability measurements on the population of all blades not exceed $\sigma^{2}=0.18 \mathrm{~mm}^{2}$. An engine inspector took a random sample of 61 fan blades from an engine. She measured each blade and found a sample variance of $0.27 \mathrm{~mm}^{2}$. Using a $0.01$ level of significance, is the inspector justified in claiming that all the engine fan blades must be replaced? Find a $90 \%$ confidence interval for the population standard deviation.

Please provide the following information.
(a) What is the level of significance? State the null and alternate hypotheses.
(b) Find the value of the chi-square statistic for the sample. What are the degrees of freedom? What assumptions are you making about the original distribution?
(c) Find or estimate the $P$ -value of the sample test statistic.
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis of independence?
(e) Interpret your conclusion in the context of the application.
(f) Find the requested confidence interval for the population variance or population standard deviation. Interpret the results in the context of the application.
Engineering: Jet Engines The fan blades on commercial jet engines must be replaced when wear on these parts indicates too much variability to pass inspection. If a single fan blade broke during operation, it could severely endanger a flight. A large engine contains thousands of fan blades, and
safety regulations require that variability measurements on the population of all blades not exceed $\sigma^{2}=0.18 \mathrm{~mm}^{2}$. An engine inspector took a random sample of 61 fan blades from an engine. She measured each blade and found a sample variance of $0.27 \mathrm{~mm}^{2}$. Using a $0.01$ level of significance, is the inspector justified in claiming that all the engine fan blades must be replaced? Find a $90 \%$ confidence interval for the population standard deviation.

Key Concepts

Level of Significance

This is the predetermined threshold for deciding whether the observed results are sufficiently extreme to reject the null hypothesis. It represents the probability of making a Type I error and is denoted as ? (for example, 0.01). In hypothesis testing, the level of significance serves as a criterion against which the p-value is compared, ensuring that the probability of a false positive remains controlled.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) usually represents the status quo or a statement of no effect, while the alternative hypothesis (Ha) represents a claim of interest that contradicts H0. In tests of variance, H0 often specifies that the population variance is equal to a given value, whereas Ha expresses that the population variance is different (or possibly, greater than) that value. Establishing these hypotheses is critical for determining the direction of the test and the decision rule.

Chi-square Test Statistic

The chi-square test statistic is used to assess the variability in a sample relative to a hypothesized population variance. It is calculated using the formula (n-1)*s²/??², where s² is the sample variance, ??² is the hypothesized variance, and n is the sample size. This statistic follows a chi-square distribution under the null hypothesis, provided that the sample comes from a normally distributed population.

Degrees of Freedom

Degrees of freedom in the context of the chi-square test for variance refer to the number of independent values that are free to vary when estimating a statistical parameter, typically calculated as n-1 for a single sample. The degrees of freedom affect the shape of the chi-square distribution and are essential in determining critical values and p-values in hypothesis tests.

P-value

The p-value quantifies the probability of observing a test statistic as extreme as, or more extreme than, the value calculated from the sample data, assuming the null hypothesis is true. It is used to determine the strength of evidence against the null hypothesis. A low p-value indicates that the observed data are unlikely under the null hypothesis, leading to its rejection.

Decision Rule in Hypothesis Testing

The decision rule in hypothesis testing involves comparing the p-value to the level of significance. If the p-value is less than ?, the null hypothesis is rejected in favor of the alternative hypothesis. Conversely, if the p-value is greater than or equal to ?, there is insufficient evidence to reject the null hypothesis. This rule helps in making a statistically informed conclusion about the population parameter under consideration.

Confidence Interval for Population Variance/Standard Deviation

A confidence interval for the population variance or standard deviation is constructed to estimate the range within which the true parameter is likely to lie with a specified level of confidence. For variance, this interval relies on the chi-square distribution using the sample variance, the sample size, and the appropriate quantiles. When the interval is computed, it provides a measure of precision and reliability for the estimated variability of a population.

Assumptions of Normality

Many statistical tests for variance, including the chi-square test, rely on the assumption that the sample comes from a normally distributed population. This assumption is crucial because the sampling distribution of the test statistic (i.e., the chi-square distribution) is derived under normality. Violations of this assumption can lead to incorrect conclusions about the population variance.

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