a manufacturer claims that the life time of a certain brand of batteries has a varaince of 5000(hours)^2. a sample of 26 has a variance of 7200(hours)^2. assuming that it is reasonable to treat these data as a random sample from a normal population, test the manufacturer's claim at alpha=0.02 level.
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Answer
To test the manufacturer's claim, we can use the chi-square test for variance. Here are the steps to calculate and interpret the test:
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The population variance is equal to the claimed variance (5000 hours^2).
- Alternative hypothesis (Ha): The population variance is not equal to the claimed variance.
Step 2: Set the significance level (alpha) to 0.02. This represents the probability of rejecting the null hypothesis when it is true.
Step 3: Calculate the test statistic:
- The test statistic for comparing variances is given by:
chi-square = (n - 1) * sample_variance / claimed_variance
In this case, n = 26 (sample size), sample_variance = 7200 (sample variance), and claimed_variance = 5000.
Step 4: Determine the critical value:
- Since the test is two-tailed, we need to find the critical values that divide the chi-square distribution into two regions, each with an alpha/2 probability.
- The degrees of freedom for the test is given by df = n - 1 = 26 - 1 = 25.
- Using a chi-square table or calculator, find the critical chi-square values at alpha/2 = 0.02/2 = 0.01 and degrees of freedom = 25.
Step 5: Compare the test statistic with the critical value:
- If the test statistic is within the critical region, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.
Step 6: Draw the conclusion:
- Based on the comparison in step 5, make a conclusion about the null hypothesis. In this case, state whether or not there is enough evidence to support the manufacturer's claim.
Note: The chi-square test assumes that the data is a random sample from a normal population, which is assumed to be reasonable in this case.
Now, let's work through the calculations:
Step 3: Calculate the test statistic:
chi-square = (n - 1) * sample_variance / claimed_variance
= (26 - 1) * 7200 / 5000
= 25 * 1.44
= 36
Step 4: Determine the critical value:
- Using a chi-square calculator or table, find the critical chi-square values at alpha/2 = 0.01 and degrees of freedom = 25. The critical value is approximately 41.344.
Step 5: Compare the test statistic with the critical value:
Since the test statistic (36) is less than the critical value (41.344), it falls within the non-critical region.
Step 6: Draw the conclusion:
Since the test statistic falls within the non-critical region, we fail to reject the null hypothesis. Therefore, there is not enough evidence to support the manufacturer's claim about the battery lifetime variance.
In conclusion, at a significance level of 0.02, the sample data does not provide sufficient evidence to support the manufacturer's claim that the variance of the battery lifetime is 5000 (hours)^2.