The mean lifetime of a sample of 100 light tubes produced by a company is found to be 1570 hours with standard deviation of 80 hours. Test the hypothesis that the mean lifetime of the tubes produced by the company is 1600 hrs
Z = (score-mean)/SD = (1600-1570)/SEm
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability. Relate to your level of significance (.05 or .01).
To test the hypothesis that the mean lifetime of the light tubes produced by the company is 1600 hours, we can use a hypothesis test using the z-test.
Here are the steps to perform the hypothesis test:
Step 1: State the null and alternative hypothesis:
- Null hypothesis (H0): The mean lifetime of the light tubes produced by the company is 1600 hours.
- Alternative hypothesis (Ha): The mean lifetime of the light tubes produced by the company is not equal to 1600 hours.
Step 2: Determine the level of significance (α):
The level of significance, denoted by α, is the maximum probability of rejecting the null hypothesis when it is actually true. It is commonly set to 0.05 or 5%.
Step 3: Calculate the test statistic:
We can use the z-test statistic formula to calculate the test statistic. The formula is:
z = (sample mean - population mean) / (standard deviation / sqrt(sample size))
In this case:
- Sample mean (x̄) = 1570 hours
- Population mean (μ) = 1600 hours
- Standard deviation (σ) = 80 hours
- Sample size (n) = 100
Plug in these values into the formula and calculate the z-score.
z = (1570 - 1600) / (80 / sqrt(100))
z = -30 / (80 / 10)
z = -30 / 8
z ≈ -3.75 (rounded to two decimal places)
Step 4: Determine the critical value:
To compare the test statistic to the critical value, we need to determine the critical value for the chosen level of significance (α). This value is obtained from the standard normal distribution table (Z-table). For a two-tailed test at α = 0.05, we divide the significance level by 2 to get α = 0.025. Looking up the critical value in the Z-table, we find it to be approximately ±1.96.
Step 5: Make a decision:
Compare the test statistic (z-score) to the critical value. If the test statistic falls outside the critical region (reject region), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
In this case, the z-score is -3.75, which is less than -1.96. Therefore, the test statistic falls in the rejection region.
Step 6: Interpret the result:
Since the test statistic falls in the rejection region, we reject the null hypothesis. There is enough evidence to suggest that the mean lifetime of the light tubes produced by the company is not 1600 hours.
Conclusion:
Based on the given data, there is sufficient evidence to conclude that the mean lifetime of the light tubes produced by the company is not 1600 hours.