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Sandpaper is rated by the coarseness of the grit on the paper. Sandpaper that is more coarse will remove material faster. Jobs that involve the final sanding of bare wood prior to painting or sanding in between coats of paint require sandpaper that is much finer. A manufacturer of sandpaper rated 220, which is used for the final preparation of bare wood, wants to make sure that the variance of the diameter of the particles in their 220 sandpaper does not exceed \(2.0\) micrometers. Fifty-one randomly selected particles are measured. The variance of the particle diameters is \(2.13\) micrometers. Assume that the distribution of particle diameter is approximately normal. a. Construct the \(95 \%\) confidence intervals for the population variance and standard deviation. b. Test at a \(2.5 \%\) significance level whether the variance of the particle diameters of all particles in 220 -rated sandpaper is greater than \(2.0\) micrometers.

Step by step solution

01

Calculate the confidence interval for the variance

To construct the 95% confidence intervals for the variance, use the chi-square distribution. The formula for the confidence interval is given by: \((n-1)s^2 / \chi^2_{(1-\alpha/2, n-1)}, (n-1)s^2 / \chi^2_{(\alpha/2, n-1)}\), where \(n\) is the sample size, \(s^2\) is the sample variance, \(\alpha\) is the level of significance, and \(\chi^2\) is the chi-square distribution. Substituting the given values: sample size \(n = 51\), sample variance \(s^2 = 2.13\), and the level of significance \(\alpha = 0.05\) yields: \((50*2.13/73.361), (50*2.13/28.416)\) = \(1.45, 3.74\) micrometers^2.

02

Calculate the confidence interval for the standard deviation

The standard deviation is just the square root of the variance. Therefore, for our interval, take the square roots of the endpoints to obtain the 95% confidence interval for the standard deviation: \(\sqrt{1.45}, \sqrt{3.74}\) = \(1.20, 1.93\) micrometers.

03

Formulation of hypotheses

For part b, formulate the null and alternative hypotheses. The null hypothesis assumes that the variance of the particle diameters is 2, denoted as \(H_0: \sigma^2 = 2\). The alternative hypothesis is that the variance is greater than 2, denoted as \(H_a: \sigma^2 > 2\). This is a one-tailed test.

04

Calculation of test statistic

Use the chi-square test statistic to test the hypothesis, which is calculated as \((n-1)s^2 / \sigma_0^2\), where \(\sigma_0^2\) is the assumed true variance under the null hypothesis. Substituting the values, we get \(50*2.13/2 = 53.25\). This chi-square statistic follows a chi-square distribution with \(n-1 = 50\) degrees of freedom.

05

Hypothesis testing

The critical value of chi-square for a 2.5% significance level with 50 degrees of freedom (one-tailed) is 70.222. Compare the test statistic with the critical value; 53.25 is less than 70.222. Therefore, we cannot reject the null hypothesis. Hence, we do not have enough evidence to say that the variance of the sandpaper particle diameters is greater than 2 micrometers.

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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 critical concept in statistical inference, particularly when dealing with variance and standard deviation of a sample. It's a non-negative, right-skewed distribution commonly used in hypothesis testing and constructing confidence intervals.
This distribution is essential when assessing if a sample's variance is statistically different from the population variance.
For example, in the context of sandpaper grading - as seen in our exercise - the chi-square test is used to examine if the variance in particle diameter exceeds a given threshold, ensuring quality control.
Remember: the shape of the chi-square distribution depends on the degrees of freedom. More degrees mean closer resemblance to a normal distribution, which impacts how we interpret results in our statistical tests.

Confidence Interval

A confidence interval provides a range where the true parameter (like variance or standard deviation) is expected to lie, given a certain level of confidence, such as 95%. It is an interval estimate, not a guarantee, indicating how confident we are that it contains the true parameter.

• To calculate confidence intervals for variance, the chi-square distribution is employed, using the formula: \[ \left(\frac{(n-1)s^2}{\chi^2_{(1-\alpha/2, n-1)}}, \frac{(n-1)s^2}{\chi^2_{(\alpha/2, n-1)}}\right) \]
• In our exercise, the variance confidence interval was calculated for sandpaper particles, providing insight into the variability of particle size. This is crucial in verifying that production standards are met.

Confidence intervals help us grasp the precision and reliability of our estimates, allowing informed decisions based on statistical evidence.

Hypothesis Testing

Hypothesis testing is a method of statistical inference used to decide if there is enough evidence to reject a hypothesis about a population parameter. It's a fundamental part of understanding statistical conclusions.

• In our sandpaper example, we set up a null hypothesis (\( H_0: \sigma^2 = 2 \)), asserting that the variance equals 2 micrometers squared.
• The alternative hypothesis (\( H_a: \sigma^2 > 2 \)) proposes the variance might be greater.

Using the chi-square distribution, we calculate a test statistic which determines whether to reject the null hypothesis. Rejecting suggests sufficient evidence that the population variance is different from the hypothesized value. In our case, the test concludes we do not reject the null hypothesis, implying that the variance might still meet the quality standard.

Variance

Variance is a measure of how much the values in a dataset differ from the mean, indicating data's dispersion or spread. In statistical terms, this tells us about the consistency within data points.
Formally, it's calculated by: \[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \] where \( x_i \) are data points, \( \bar{x} \) is the mean, and \( n \) is the sample size.
In our context, variance checks if sandpaper particles' diameters are consistent or widely different - crucial for maintaining the grading of sandpaper. If the variance is too high, it might mean some particles are too large or small, affecting the sandpaper's performance.
Therefore, maintaining control over variance ensures products meet the desired quality.

Standard Deviation

Standard deviation is the square root of variance, providing a more intuitive measure of spread by expressing it in the same unit as the data. It's a significant indicator of variability.
In formula terms, standard deviation is written as: \[ s = \sqrt{s^2} \]
This statistic augments variance by translating abstract variability into values comparable to the mean. For the sandpaper exercise, after determining the variance confidence interval, calculating the standard deviation gives further practical insights into size fluctuations.
Lower standard deviation relative to the mean suggests more data clustering around the average, crucial for predicting how sandpaper will perform during its intended use.