A machine produces disks with mean thickness 2 mm . To test...

A machine produces disks with mean thickness 2 mm . To test the machine after undergoing routine maintenance, 10 sample disks are selected and the sample mean of the thickness is found to be 2.2 mm and the sample variance is found to be $0.04 \mathrm{~mm}^2$. (a) Find a test to determine if the machine is working properly for $\alpha=1 \%$; $\alpha=5 \%$. (b) Find the $p$-value of the observation.

A machine produces disks with mean thickness 2 mm . To test the machine after undergoing routine maintenance, 10 sample disks are selected and the sample mean of the thickness is found to be 2.2 mm and the sample variance is found to be $0.04 \mathrm{~mm}^2$.
(a) Find a test to determine if the machine is working properly for $\alpha=1 \%$; $\alpha=5 \%$.
(b) Find the $p$-value of the observation.

Key Concepts

Degrees of Freedom

Degrees of freedom (df) refer to the number of independent values or quantities that can be assigned to a statistical distribution. In the context of a t-test, df is typically calculated as the sample size minus one, and influences the shape of the t-distribution used to determine p-values and critical values.

p-value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic as extreme or more extreme than the one observed. It provides a measure of the evidence against the null hypothesis, with smaller p-values indicating stronger evidence for the alternative hypothesis.

Significance Level

The significance level, denoted as alpha (?), represents the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. Testing at different levels (such as 1% or 5%) determines how stringent the test is; lower values demand stronger evidence against the null hypothesis before it can be rejected.

Null and Alternative Hypotheses

These hypotheses form the basis of the test. The null hypothesis (H0) typically states that there is no change or difference (for example, that the machine produces disks with a mean thickness of 2 mm), while the alternative hypothesis (H1) indicates a deviation from this standard (that the mean thickness is not 2 mm). Framing these properly is essential for drawing correct conclusions from the test.

Hypothesis Testing

This concept involves evaluating a claim about a population parameter based on sample data. It consists of setting up a null hypothesis (that represents the standard or status quo condition) and an alternative hypothesis (what you suspect might be true), then using sample statistics to assess whether the observed data provide sufficient evidence to reject the null hypothesis.

Test Statistic (t-test)

The t-test is used when the sample size is small and the population variance is unknown. In this framework, the test statistic is calculated using the difference between the sample mean and the claimed population mean, scaled by the standard error of the sample mean. This statistic follows a t-distribution with a specified number of degrees of freedom.

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