91Ó°ÊÓ

A company manufactures ball bearings that are supplied to other companies. The machine that is used to manufacture these ball bearings produces them with a variance of diameters of \(.025\) square millimeter or less. The quality control officer takes a sample of such ball bearings quite often and checks, using confidence intervals and tests of hypotheses, whether or not the variance of these bearings is within \(.025\) square millimeter. If it is not, the machine is stopped and adjusted. A recently taken random sample of 23 ball bearings gave a variance of the diameters equal to \(.034\) square millimeter. a. Using a \(5 \%\) significance level, can you conclude that the machine needs an adjustment? Assume that the diameters of all ball bearings have a normal distribution. b. Construct a \(95 \%\) confidence interval for the population variance.

Step by step solution

01

Hypothesis Testing

Let's start by stating the null and alternate hypotheses. The null hypothesis \(H_0\) is that the variance is \(\leq 0.025\) and the alternate hypothesis \(H_a\) is that the variance is \(> 0.025\). The Chi-Square test statistic (X) is computed from the sample data by using the formula \[X = \frac{(n - 1)S^2}{\sigma^2}\], where \(n\) is the sample size (23), \(S^2\) is the sample variance (0.034), and \(\sigma^2\) is the claimed population variance (0.025). Substituting these values gives \(X = 31.28\). We look at the Chi-Square table for \(n - 1 = 22\) degrees of freedom and find the critical value corresponding to the \(5\%\) level. We can see that critical value is \(33.92\). Since \(31.28 < 33.92\), we fail to reject the null hypothesis. Therefore, there isn't sufficient evidence that the machine needs adjustment at the \(5\%\) significance level.

02

Confidence Interval For Variance

To calculate a \(95\%\) confidence interval for the population variance, we will again use the Chi-Square distribution. The formula for the \(95\%\) confidence interval for a population variance is \[\left[\frac{(n - 1)S^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n - 1)S^2}{\chi^2_{1-\alpha/2, n-1}}\right]\], where \(\chi^2\) denotes the Chi-Square critical values at the given level of significance (\(5\%\)) with \(n - 1\) degrees of freedom. Looking in the Chi-Square table, the critical values are \(\chi^2_{0.025, 22} = 11.18\) and \(\chi^2_{0.975, 22} = 39.36\). Substituting these values in our formula results in the confidence interval for variance : \[\left[0.022, 0.068\right]\] square millimeter.

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Key Concepts

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

Chi-Square Distribution

The Chi-Square distribution is a vital concept in statistics, particularly when it comes to hypothesis testing and confidence intervals relating to variance. This distribution is used when dealing with datasets that have certain characteristics, like non-negative data. It's especially handy for working with variance because of its relationship with sums of squared deviations.

In hypothesis testing, the Chi-Square distribution helps determine if the observed variance significantly differs from the expected variance under the null hypothesis.

• We compute the Chi-Square statistic using the formula: \[X = \frac{(n - 1)S^2}{\sigma^2}\]where \(n\) is sample size, \(S^2\) is sample variance, and \(\sigma^2\) is the population variance.
• By comparing this statistic to the critical value from the Chi-Square distribution table, we can decide to reject or fail to reject the null hypothesis.
• This distribution is characterized by degrees of freedom, typically \(n-1\) for sample variance, and it skews to the right with its shape depending on these degrees.

Understanding the Chi-Square distribution is crucial because it provides a framework for the reliability of variance estimates and testing manufacturing consistency.

Confidence Intervals

Confidence intervals provide a range of values within which we expect the true population parameter, like variance, to lie. They are crucial for making informed decisions about population characteristics with a known level of certainty.

For variance, the confidence interval calculation utilizes the Chi-Square distribution. The formula used to calculate the interval is:\[\left[\frac{(n - 1)S^2}{\chi^2_{\alpha/2, n-1}}, \frac{(n - 1)S^2}{\chi^2_{1-\alpha/2, n-1}}\right]\]where:

• \(\alpha\) represents the significance level, e.g., 0.05 for a 95% confidence level.
• \(\chi^2\) values are critical Chi-Square values for your chosen confidence level.
• The result is a range providing the boundaries within which we are confident the accepted variance falls.

Confidence intervals help not only in hypothesis testing checks but also in quality control contexts, like with the diameter of ball bearings, to ensure consistency.

Variance

Variance measures the spread or variability of a set of data points around their mean. In simple terms, it gives us an idea of how much the numbers in the dataset differ from the mean of the dataset.

In manufacturing settings, like the production of ball bearings, variance checks help maintain quality consistency.

• It is calculated using the formula, \[S^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2\]where \(x_i\) is each data point, \(\bar{x}\) is the mean, and \(n\) is the number of observations.
• A smaller variance implies that the data points are closely packed around the mean, while a larger variance indicates a wider spread.
• It is crucial in deciding if production methods need adjustments, as indicated if the variance exceeds certain thresholds.

Understanding variance is necessary for recognizing variations in process control which can impact overall production quality.

Sample Size

Sample size, typically denoted by \(n\), is the number of observations or data points considered in a statistical sample. It plays a crucial role in statistical analysis, especially when we aim for significant results regarding population parameters.

In the context of hypothesis testing and confidence intervals for variance:

• A larger sample size generally gives a more accurate estimate of the population variance.
• It also influences the degrees of freedom in the Chi-Square distribution, given by \(n-1\) for variance testing.
• With increased sample size, the margin of error in confidence intervals reduces, making the interval narrower.

Choosing an appropriate sample size ensures that the statistical findings have sufficient power to provide conclusive evidence about the population, offering more reliable results for decision-making in manufacturing processes like those of ball bearings.