A machine used for packaging seedless golden raisins is set so that their standard deviation in the weight of raisins packaged per box is 0.25 ounce. The operations manager wishes to test the machine setting and selects a sample of 30 consecutive raisin packages filled during the production process.
Weights
15.2
15
14.3
15.3
15.2
14.4
15.1
15.4
15.5
15.7
15.6
15.4
15.3
15.7
15.2
15
15.4
15.5
15.1
15.3
15.6
14.3
14.9
15.1
14.6
14.8
15.3
14.5
14.6
15.1
At the 0.05 level of significance, is there evidence that the population standard deviation differs from 0.25 ounces?
What assumptions are made in order to perform this test?
First establish a null hypothesis
H0 : population σ² = 0.25²
n = 30
Calculate the sample variance, s.
calculate the Χ² statistic
Χ² = (n-1)*s²/σ²
Look up the Χ² table for 29 degrees of freedom and 5 and 95% levels of confidence.
If the calculated Χ² falls between these limits, the null hypothesis is not rejected.
To determine if there is evidence that the population standard deviation differs from 0.25 ounces, we need to perform a hypothesis test. Specifically, we are testing the null hypothesis (H0) that the population standard deviation is equal to 0.25 ounces against the alternative hypothesis (Ha) that the population standard deviation is different from 0.25 ounces.
To perform this test, we can use the chi-square distribution with (n-1) degrees of freedom, where n is the sample size.
Here are the steps to conduct the hypothesis test:
1. Calculate the sample standard deviation (s) of the raisin weights in the given sample. In this case, the sample size is 30.
2. Formulate the null and alternative hypotheses:
- Null hypothesis (H0): The population standard deviation is equal to 0.25 ounces.
- Alternative hypothesis (Ha): The population standard deviation is different from 0.25 ounces.
3. Determine the critical values for the test. Since we are conducting a two-tailed test at a significance level of 0.05, the critical values will be obtained from the chi-square distribution based on (n-1) degrees of freedom and the desired significance level.
4. Calculate the test statistic (chi-square test statistic) using the formula:
chi-square = (n - 1) * (s^2) / (0.25^2)
- 'n' is the sample size.
- 's' is the sample standard deviation.
5. Compare the calculated test statistic with the critical values:
- If the calculated test statistic is greater than the critical value, we reject the null hypothesis and conclude that the population standard deviation differs from 0.25 ounces.
- If the calculated test statistic falls within the critical region, we fail to reject the null hypothesis and do not have sufficient evidence to conclude that the population standard deviation differs from 0.25 ounces.
Assumptions for this test:
1. The raisin weights are a random sample from the population.
2. The raisin weights are normally distributed.
3. The sample standard deviation (s) can be considered an unbiased estimator of the population standard deviation (σ).
By following these steps and conducting the hypothesis test, you can determine if there is evidence that the population standard deviation differs from 0.25 ounces at the 0.05 level of significance.