1. An assembly line produces widgets with a mean weight of 10 and a standard deviation of 0.200. A new process supposedly will produce widgets with the same mean and a smaller standard deviation. A sample of 20 widgets produced by the new method has a sample standard deviation of 0.126. At a significance level of 10%, can we conclude that the new process is less variable than the old? State appropriate null and alternative hypotheses.
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To determine if the new process is less variable than the old process, we can conduct a hypothesis test.
Null hypothesis (H0): The standard deviation of the new process is the same as the standard deviation of the old process.
Alternative hypothesis (HA): The standard deviation of the new process is smaller than the standard deviation of the old process.
Now, let's perform the hypothesis test using the given information:
1. Set up the test statistic:
- Sample size: n = 20
- Sample standard deviation: s = 0.126
- Population standard deviation: σ (old process) = 0.200
2. Calculate the test statistic:
- Test statistic (chi-square): χ^2 = (n - 1) * (s / σ)^2
- Plugging in the values: χ^2 = (20 - 1) * (0.126 / 0.200)^2
3. Determine the critical value:
- The degrees of freedom for a chi-square test with (n - 1) degrees of freedom and a significance level of 10% are (n - 1) = 19.
4. Compare the test statistic to the critical value:
- If the test statistic is less than the critical value, we reject the null hypothesis and conclude that the new process is less variable than the old process. Otherwise, we fail to reject the null hypothesis.
5. Calculate the test statistic:
- χ^2 = (19) * (0.63)^2
- χ^2 = 7.5633
6. Determine the critical value:
- Using a chi-square distribution table or calculator, the critical value for a chi-square test with 19 degrees of freedom and a significance level of 10% is approximately 28.869.
7. Compare the test statistic to the critical value:
- 7.5633 < 28.869
Since the test statistic (7.5633) is less than the critical value (28.869), we reject the null hypothesis. Therefore, we can conclude that at a significance level of 10%, the new process is less variable than the old process.
To determine if we can conclude that the new process is less variable than the old process, we can perform a hypothesis test.
First, let's state the null and alternative hypotheses:
Null hypothesis (H0): The new process has the same or greater standard deviation than the old process.
Alternative hypothesis (Ha): The new process has a smaller standard deviation than the old process.
Next, we will determine the critical value or the test statistic to compare with the observed sample statistic.
Since we are comparing the variances of two normal populations, we can use the F-test. The test statistic for the F-test is calculated as the ratio of the sample variances, which follows an F-distribution.
To perform the F-test, we need to calculate the F-statistic using the following formula:
F = (s1^2) / (s2^2)
Where s1^2 is the sample variance of the new process, and s2^2 is the sample variance of the old process.
In our case, the sample standard deviation of the new process is given as 0.126, so we can calculate the sample variance as follows:
s1^2 = (0.126)^2 = 0.015876
The population standard deviation of the old process is given as 0.200, so we can calculate the population variance as follows:
s2^2 = (0.200)^2 = 0.040000
Now we can calculate the F-statistic:
F = 0.015876 / 0.040000 = 0.3969
Note: The F-statistic follows an F-distribution with degrees of freedom (df1, df2), where df1 is the degrees of freedom for the numerator (sample variance of the new process) and df2 is the degrees of freedom for the denominator (sample variance of the old process). In this case, df1 = n1 - 1 = 20 - 1 = 19, and df2 = n2 - 1 = unknown.
To find the critical value of the F-distribution at a significance level of 10%, we need to use a statistical table or statistical software. The critical value will depend on the degrees of freedom.
Once we have the critical value, we can compare it with the F-statistic. If the F-statistic is less than the critical value, we reject the null hypothesis and conclude that the new process is less variable than the old process. If the F-statistic is greater than or equal to the critical value, we fail to reject the null hypothesis, and we do not have enough evidence to conclude that the new process is less variable than the old process.
Please refer to a statistical table or use statistical software to find the critical value for the F-distribution with the appropriate degrees of freedom.