Your muffler factory claims to manufacture mufflers with a lifespan of more than 10000 miles of usage. A consumer group tests this claim at the 95 percent significance level, and finds that a sample of 64 mufflers have a mean lifespan of 10002 miles, with a standard deviation of 10 miles. Test the following alternative hypothesis using this data, and interpret the results:

(a) manufacturer's hypothesis : Ha : mean u is greater than 10000
(b) consumer group's hypothesis : Ha: mean u is less than 10000
(c) if the manufacturer wanted to state that the survey proved their claim to be true, what should mean x have been?
(d) if the consumer group wanted to state that the survey proved the manufacturer's claim to be false, what should mean x have been?

To test the alternative hypotheses and interpret the results, we can perform a one-sample t-test. Here's how we can do that:

(a) Manufacturer's hypothesis: Ha: μ > 10000
We want to test if the true mean lifespan of mufflers is greater than 10000 miles.

(b) Consumer group's hypothesis: Ha: μ < 10000
We want to test if the true mean lifespan of mufflers is less than 10000 miles.

To compute the test statistics, we can calculate the t-value using the formula:

t = (x̄ - μ) / (s / sqrt(n))

Where:
x̄ = sample mean = 10002
μ = assumed population mean = 10000
s = sample standard deviation = 10
n = sample size = 64

Let's calculate the t-value for each hypothesis:

(a) Manufacturer's hypothesis:
t = (10002 - 10000) / (10 / sqrt(64)) = 2 / (10 / 8) = 1.6

(b) Consumer group's hypothesis:
t = (10002 - 10000) / (10 / sqrt(64)) = 2 / (10 / 8) = 1.6

Next, we need to find the critical t-value at the 95% significance level with degrees of freedom (df) equal to n-1 = 63. We can use a t-table or a statistical software to find this value.

Let's assume the critical t-value for the one-tailed test is 1.65.

Now we can compare our calculated t-value with the critical t-value:

(a) Manufacturer's hypothesis: 1.6 < 1.65
Since the calculated t-value is less than the critical t-value, we fail to reject the null hypothesis. There is not enough evidence to support the claim that the mean lifespan of mufflers is greater than 10000 miles.

(b) Consumer group's hypothesis: 1.6 < 1.65
Since the calculated t-value is less than the critical t-value, we also fail to reject the null hypothesis. There is not enough evidence to support the claim that the mean lifespan of mufflers is less than 10000 miles.

(c) If the manufacturer wanted to state that the survey proved their claim to be true, the sample mean x̄ would have needed to be statistically significantly greater than 10000 miles. This means x̄ would have needed to be larger than the critical t-value multiplied by (s / sqrt(n)) plus 10000.

(d) If the consumer group wanted to state that the survey proved the manufacturer's claim to be false, the sample mean x̄ would have needed to be statistically significantly less than 10000 miles. This means x̄ would have needed to be smaller than the critical t-value multiplied by (s / sqrt(n)) plus 10000.

Note: The critical t-value used in this explanation is just an example. In practice, you should use the exact critical t-value based on the degrees of freedom and significance level of your specific test.

To test the alternative hypotheses, we need to perform a one-sample t-test. Let's go through the steps for each hypothesis:

(a) Manufacturer's hypothesis: Ha: μ > 10000
To test this hypothesis, we will perform a one-tailed t-test, where the null hypothesis is that the mean lifespan is equal to or less than 10000 miles (Ho: μ ≤ 10000). The alternative hypothesis Ha states that the mean lifespan is greater than 10000 miles.

1. Calculate the test statistic:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean (10002 miles), μ is the hypothesized mean (10000 miles), s is the standard deviation (10 miles), and n is the sample size (64).

t = (10002 - 10000) / (10 / √64)
t = 2 / (10 / 8)
t = 1.6

2. Determine the critical value:
Since the significance level is 95%, we need to find the critical value for a one-tailed test at the 95th percentile. We can use a t-table or a t-distribution calculator to find the critical t-value with (n-1) degrees of freedom (63 degrees of freedom in this case). Let's assume the critical t-value is 1.669.

3. Compare the test statistic with the critical value:
If the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Since t = 1.6 < 1.669, we fail to reject the null hypothesis.

Interpretation:
Based on the data, there is not enough evidence to support the claim that the mean lifespan of the mufflers is greater than 10000 miles at the 95% significance level.

(b) Consumer group's hypothesis: Ha: μ < 10000
To test this hypothesis, we will perform a one-tailed t-test, where the null hypothesis is that the mean lifespan is equal to or greater than 10000 miles (Ho: μ ≥ 10000). The alternative hypothesis Ha states that the mean lifespan is less than 10000 miles.

1. Calculate the test statistic:
t = (x̄ - μ) / (s / √n)
where x̄ is the sample mean (10002 miles), μ is the hypothesized mean (10000 miles), s is the standard deviation (10 miles), and n is the sample size (64).

t = (10002 - 10000) / (10 / √64)
t = 2 / (10 / 8)
t = 1.6

2. Determine the critical value:
Since the significance level is 95%, we need to find the critical value for a one-tailed test at the 5th percentile. We can use a t-table or a t-distribution calculator to find the critical t-value with (n-1) degrees of freedom (63 degrees of freedom in this case). Let's assume the critical t-value is -1.669.

3. Compare the test statistic with the critical value:
If the test statistic is less than the critical value, we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Since t = 1.6 > -1.669, we reject the null hypothesis.

Interpretation:
Based on the data, there is enough evidence to support the claim that the mean lifespan of the mufflers is less than 10000 miles at the 95% significance level.

(c) If the manufacturer wanted to state that the survey proved their claim to be true, what should mean x have been?
If the manufacturer wanted to prove their claim to be true, the sample mean (x̄) should have been greater than or equal to 10000 miles. In this case, the sample mean (x̄) is 10002 miles, which is indeed greater than 10000 miles.

(d) If the consumer group wanted to state that the survey proved the manufacturer's claim to be false, what should mean x have been?
If the consumer group wanted to prove the manufacturer's claim to be false, the sample mean (x̄) should have been less than 10000 miles. In this case, the sample mean (x̄) is 10002 miles, which is not less than 10000 miles.