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Magazine Chapter 9: Problem 137
In the past the standard deviation of weights of certain 32.0 -oz packages filled by a machine was 0.25 oz. A random sample of 20 packages showed a standard deviation of 0.35 oz. Is the apparent increase in variability significant at the 0.10 level of significance? Assume package weight is normally distributed. a. Solve using the \(p\) -value approach. b. Solve using the classical approach.
Short Answer
Expert verified
According to the \(p\)-value approach, there is a significant increase in variability. However, the classical approach doesn't support this conclusion. Therefore, the results are inconclusive.
Step by step solution
01
State the hypotheses
The null hypothesis \(H_0\) is that the variance \( \sigma^2 = 0.0625\). The alternative hypothesis \(H_1\) is that the variance \( \sigma^2 > 0.0625\)
02
Find the test statistic
Once the hypotheses are defined, you need to calculate the test statistic. For testing variance, the Chi-square statistic is used. This can be done using the formula \( \frac{(n-1)s^2}{\sigma_0^2}\), where \(n\) is the sample size, \(s^2\) is the sample variance, and \(\sigma_0^2\) is the speculated variance. Here, \(n=20\), \(s^2=0.35^2\) and \(\sigma_0^2=0.25^2\). Substituting these values gives a test statistic of 10.08.
03
Find the p-value
Given the calculated test statistic and the sample size, you can find the p-value by referring to a chi-square distribution table. The degree of freedom is \(n-1 = 20-1 = 19\). The p-value in this case is the area to the right of the test statistic, which is less than 0.10.
04
Make Decision - p-value approach
By comparing the p-value and the level of significance (0.10), it can be seen that the p-value is less than the level of significance. Hence, the null hypothesis is rejected by the p-value approach. Which means, there is a significant increase in variability of the packages' weight.
05
Find the critical value - Classical approach
To solve this using the classical approach, you need to find the critical value from a chi-square distribution table. For a 0.10 level of significance and degree of freedom 19, the critical value is 30.144.
06
Make Decision - Classical approach
The test statistic is compared with the critical value. If the test statistic is greater than the critical value, then the null hypothesis is rejected. In this case, 10.08 is not greater than 30.144. Hence, the null hypothesis is not rejected by the classical approach. There is not enough evidence to conclude that there is a significant increase in variability of the packages' weight.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Hypothesis Testing
Hypothesis testing is a method used in statistics to decide whether there is enough evidence to reject a proposed assumption about a population parameter. The process involves making a conjecture (the null hypothesis) that a population parameter, like the mean or variance, is equal to a specific value. The alternative hypothesis, on the other hand, represents what we want to prove - that the parameter is different from the value stated in the null hypothesis.
Sample Variance
Sample variance is a measure of how much a set of observations varies from their mean. It is calculated by taking the squared differences from the mean for each data point, summing them up, and dividing by the number of observations minus one. Higher variance indicates that the data points are more spread out from the mean. In the exercise, we're concerned not just with the mean but with understanding whether there's been a significant change in variability of package weights.
Level of Significance
The level of significance, denoted by \(\alpha\), is the probability of rejecting the null hypothesis when it is actually true (a Type I error). It's a threshold that dictates how sure we need to be before we can reject the null hypothesis. Common significance levels are 0.10, 0.05, and 0.01. In the exercise provided, a 0.10 level of significance is used, meaning we would be willing to accept a 10% chance of making a Type I error.
Degrees of Freedom
Degrees of freedom refer to the number of values in a calculation that are free to vary. They are a crucial component in various statistical calculations, particularly those related to variance and chi-square tests. For a sample variance, the degrees of freedom are \(n - 1\), where \(n\) is the sample size. In our exercise, the sample size of 20 packages reduces the degrees of freedom to 19 for the chi-square test.
P-value Approach
The p-value is the probability of obtaining test results at least as extreme as the ones observed during the test, assuming that the null hypothesis is correct. A smaller p-value means that there is stronger evidence in favor of the alternative hypothesis. In the given exercise, the p-value approach involves comparing the p-value against the level of significance. If it's lower, the null hypothesis is rejected, suggesting that the increase in variability is statistically significant.
Classical Approach
The classical approach, also known as the critical value approach, involves comparing the test statistic to a critical value from a statistical distribution. If the value of the test statistic is more extreme than the critical value, the null hypothesis is rejected. In this approach, one does not calculate a p-value but instead uses pre-determined cut-off points (critical values) which are based on the level of significance and degrees of freedom.
Null Hypothesis
The null hypothesis (\(H_0\)) is a statement that indicates no effect or no difference. It is a hypothesis that is presumed to be true until statistical evidence in the form of a hypothesis test indicates otherwise. In the exercise's context, the null hypothesis posits that there has been no change in the variance of the weights of the packages.
Alternative Hypothesis
The alternative hypothesis (\(H_1\) or \(_a\)), suggests that there is an effect or a difference. It is what researchers aim to support. For the packages' weight variability, the alternative hypothesis claims that there has been an increase in variance from its previous value.
Test Statistic
A test statistic is a standardized value that is calculated from sample data during a hypothesis test. It's used to decide whether to reject the null hypothesis. It represents how far our sample statistic is from the null hypothesis value of the population parameter, standardized by the standard error of the statistic. In the chi-square test for variance, the test statistic helps in determining whether the observed variance is significantly different from the expected variance under the null hypothesis.
