National Packing Materials Company claims its X20 box can hold loads up to at least 80 pounds. Smith Widget Company has been using the boxes for one year and feels the boxes fail at lighter loads. Hearing the complaints from Smith, the lead engineer for National Packaging decides to test the company’s claim that it can hold at least 80 pounds. He decides to text 64 boxes at the .05 level of significance. He finds a mean of 78 pounds with a standard deviation of 5 pounds.
What is the critical value for this test?
What is the value of the test statistic?
What is the Null hypothesis?
What is the alternate hypothesis?
What is the standard error of the mean?
Should the Null hypothesis be accepted or rejected?
To answer these questions, we need to perform a hypothesis test. Here's how we can find the answers step by step:
1. Critical value (Z-score):
The critical value can be found using the significance level and the standard normal distribution table. In this case, the significance level is 0.05, which corresponds to a 95% confidence level (1 - 0.05 = 0.95). For a two-tailed test, we divide the significance level by 2 to get 0.025. Looking up this value in the standard normal distribution table, we find the critical value to be approximately 1.96.
2. Test statistic (Z-score):
The test statistic in this case is the Z-score. It is calculated by subtracting the hypothesized mean (80 pounds) from the sample mean (78 pounds) and dividing it by the standard deviation (5 pounds). So, (78 - 80) / 5 = -0.4.
3. Null hypothesis:
The null hypothesis (H0) for this test is that the mean weight the X20 boxes can hold is equal to or greater than 80 pounds.
4. Alternate hypothesis:
The alternate hypothesis (Ha) for this test is that the mean weight the X20 boxes can hold is less than 80 pounds.
5. Standard error of the mean:
The standard error of the mean (SE) is calculated by dividing the standard deviation by the square root of the sample size. In this case, the standard deviation is 5 pounds, and the sample size is 64 boxes. So, SE = 5 / √64 = 5 / 8 = 0.625.
6. Null hypothesis acceptance or rejection:
To determine whether the null hypothesis should be accepted or rejected, we compare the test statistic (-0.4) to the critical value (1.96). If the test statistic falls in the critical region (outside the ±1.96 range), we reject the null hypothesis. Otherwise, we accept it. In this case, -0.4 is not in the critical region, so we fail to reject the null hypothesis.
To summarize:
- The critical value for this test is approximately 1.96.
- The test statistic (Z-score) is -0.4.
- The null hypothesis is that the mean weight the X20 boxes can hold is equal to or greater than 80 pounds.
- The alternate hypothesis is that the mean weight the X20 boxes can hold is less than 80 pounds.
- The standard error of the mean is 0.625.
- The null hypothesis should be accepted.