Problem 58 - An Introduction to Mathematical Statistics and Its Applications

Question

When working properly, the amounts of cement that a filling machine puts into \(25-\mathrm{kg}\) bags have a standard deviation \((\sigma)\) of \(1.0 \mathrm{~kg}\). In the next column are the weights recorded for thirty bags selected at random from a day's production. Test \(H_{0}: \sigma^{2}=1\) versus \(H_{1}: \sigma^{2}>1\) using the \(\alpha=0.05\) level of significance. Assume that the weights are normally distributed. Use the following sums: $$ \sum_{i=1}^{30} y_{i}=758.62 \text { and } \sum_{i=1}^{30} y_{i}^{2}=19,195.7938 $$

Step-by-Step Solution

Answer

Carry out the calculations in the steps: compute the sample variance \(s^2\) from the given sums, calculate the test statistic \(\chi^2\), compare the test statistic with the critical value corresponding to \(\alpha = 0.05\) and \(n - 1 = 29\) degrees of freedom. If the test statistic exceeds the critical value, the null hypothesis can be rejected.

Step 1: Compute Sample Variance

First we need to calculate the sample variance. The formula for sample variance is given by \(s^2 = \frac{\sum y_i^2 - (\sum y_i)^2 / n}{n - 1}\). Replacing the provided sums into the equation, the sample variance will be \(s^2 = \frac{19195.7938 - (758.62)^2 / 30}{30 - 1}\).

Step 2: Compute Test Statistic

Next, we need to compute the test statistic, which follows a chi-square distribution. The formula for the test statistic is \(\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}\), where \(s^2\) is the sample variance, \(n\) is the sample size and \(\sigma_0^2\) is the variance under the null hypothesis. Hence, our test statistic will be \(\chi^2 = \frac{(30 - 1)s^2}{1}\).

Step 3: Determine Critical Value and Make Decision

For a chi-square distribution with \(n - 1 = 29\) degrees of freedom and \(\alpha = 0.05\), the critical value can be looked up in a chi-square distribution table or calculated using a statistical software. If our computed test statistic exceeds the critical value, we can reject the null hypothesis.

Key Concepts

Chi-Square Test

When we want to test assumptions about the variance of a distribution, the chi-square test is often our go-to tool. Think of it as a way to check whether observed data differs significantly from what we expected under the null hypothesis. Here, we use it to test if the variance of cement bag weights is greater than 1.
To carry out the test, follow these steps:

• Calculate the sample variance from your data.
• Use the sample variance to compute the test statistic, which falls under a chi-square distribution.
• Compare this test statistic to a critical value from the chi-square distribution table based on your level of significance and degrees of freedom.

In our example, if the computed statistic is greater than the critical value, it suggests that the variability in the cement bag weights is more than expected, leading us to reject the null hypothesis.

Sample Variance

Sample variance measures how much the weights of the cement bags in our sample spread out from their average weight. Calculating sample variance involves a few clear steps:

• First, find the mean of the data set. This is the total of all measured weights divided by the number of measurements.
• Subtract each weight from this mean, square the result, and then add these squares together.
• Finally, divide by the number of observations minus one (this is the degrees of freedom).

In our calculation, the variance helps us understand the spread of bag weights and serves as a key component in the larger chi-square test. It's crucial because it provides insight into the variability within our sample, helping us verify our hypothesis about the population variance.

Level of Significance

The level of significance, often denoted by \( \alpha \), defines how strict we will be in our statistical testing. It's like setting the rules for accepting risk when making conclusions. The usual levels are 0.05, 0.01, etc. A 0.05 level of significance, like in our exercise, implies that we are 95% confident in our result—meaning there's a 5% risk of wrongly rejecting the null hypothesis.
In practical terms, the level of significance affects the threshold for decision-making in hypothesis testing. A higher \( \alpha \) implies a greater tolerance for risk, potentially leading to more rejections of the null hypothesis. On the other hand, a lower \( \alpha \) makes us more conservative, reducing false positives.
In selecting \( \alpha \), it's crucial to balance between Type I and Type II errors, aligning the test's stringency with the real-world implications of making an error in hypothesis testing.