An electronics company advertises that the battery life of its new smart phone with normal usage averages 72 hours. The company says this is one of the desirable characteristics of the phone which makes the phone a better choice compared to competitor phones. Recently, the company has received an unusually large number of complaints regarding the battery life. Many customers indicated the battery needed to be recharged long before the advertised average life. To check if the average battery life equals 72 hours, the company randomly selected 36 phones and tested the battery life. The sample produced a mean battery life of 69.5 hours with a standard deviation of 5 hours. Answer the following questions:

a. Perform the appropriate test to determine if the average battery life differs from the advertised life of 72 hours. Use the test statistic method to perform the test. Alpha for the test is .05. Be sure to include all required steps of the hypothesis test procedure as outlined and performed in class.

b. Compute a 95% confidence interval estimate for the mean battery life. How does your confidence interval support the conclusion of your hypothesis test?

a. To determine if the average battery life differs from the advertised life of 72 hours, we can perform a one-sample t-test.

Step 1: State the null and alternative hypotheses:
Null hypothesis (H0): The average battery life equals 72 hours.
Alternative hypothesis (Ha): The average battery life differs from 72 hours.

Step 2: Determine the significance level (alpha=0.05).

Step 3: Calculate the test statistic:
In this case, we can use the t-test because we have the sample mean, sample standard deviation, and the population mean for comparison. The formula for the t-test statistic is:

t = (sample mean - population mean) / (sample standard deviation / sqrt(sample size))

Given:
Sample mean (x̄) = 69.5 hours
Population mean (μ) = 72 hours
Sample standard deviation (σ) = 5 hours
Sample size (n) = 36

t = (69.5 - 72) / (5 / sqrt(36))

Step 4: Calculate the p-value associated with the t-test statistic:
Using the t-distribution table or a statistical calculator, we can find the p-value associated with the calculated t-value.

Step 5: Make a decision:
If the p-value is less than the significance level (0.05), we reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

b. To compute a 95% confidence interval estimate for the mean battery life, we can use the formula:

Confidence interval = sample mean ± (critical value * standard error)

The critical value depends on the desired confidence level (95%). Since we have a large sample size (n > 30), we can use the z-distribution and find the critical value associated with a 95% confidence level (z = 1.96).

Standard error = sample standard deviation / sqrt(sample size)

Using the given values:
Sample mean (x̄) = 69.5 hours
Sample standard deviation (s) = 5 hours
Sample size (n) = 36

Standard error = 5 / sqrt(36)

The confidence interval can be calculated as:
Confidence interval = 69.5 ± (1.96 * (5 / sqrt(36)))

The interpretation of the confidence interval will be discussed after performing the hypothesis test.

The results of the hypothesis test and the confidence interval will provide us with information on whether the average battery life of the phone differs from the advertised life of 72 hours and the range of plausible values for the population mean.

a. To determine if the average battery life differs from the advertised life of 72 hours, we can perform a hypothesis test using the test statistic method. The steps involved in this test are as follows:

Step 1: State the null hypothesis (H0) and the alternative hypothesis (Ha):
H0: The average battery life equals 72 hours.
Ha: The average battery life differs from 72 hours.

Step 2: Select the appropriate test statistic and significance level (alpha):
Since the sample size is relatively large (n > 30), we can use the z-test statistic. The significance level (alpha) is given as 0.05, which means we want a 95% confidence level.

Step 3: Formulate the decision rule:
Since the alternative hypothesis is two-sided (average battery life differs from 72 hours), we can perform a two-tailed test. With an alpha level of 0.05, we divide it by 2 to get 0.025 for each tail. We will reject the null hypothesis if the test statistic falls outside the critical value for the chosen alpha level.

Step 4: Calculate the test statistic:
The test statistic for a one-sample z-test is calculated as (sample mean - population mean) / (standard deviation / sqrt(sample size)). In this case:
Mean (sample) = 69.5
Mean (population) = 72
Standard deviation = 5
Sample size = 36

Calculating the test statistic:
z = (69.5 - 72) / (5 / sqrt(36))
z = -2.5 / (5 / 6)
z = -2.5 * (6 / 5)
z = -3

Step 5: Determine the critical value:
Since this is a two-tailed test, we need to determine the critical values for z at alpha/2 = 0.025 significance level.

Using a standard normal distribution table, the critical values are approximately -1.96 and 1.96.

Step 6: Make a decision:
Since the test statistic (-3) is less than the critical value (-1.96), we reject the null hypothesis.

b. Now, let's compute the 95% confidence interval estimate for the mean battery life:

Formula for confidence interval: sample mean ± (critical value * standard deviation / sqrt(sample size))

Using the same values as above:
Sample mean = 69.5
Standard deviation = 5
Sample size = 36

Critical value for a 95% confidence level (two-tailed) is approximately 1.96.

Confidence Interval = 69.5 ± (1.96 * 5 / sqrt(36))

Calculating the confidence interval:
Confidence Interval = 69.5 ± (1.96 * 5 / 6)
Confidence Interval = 69.5 ± (1.96 * 0.833)
Confidence Interval = 69.5 ± 1.629

Using these calculations, we find that the lower bound of the confidence interval is 69.5 - 1.629 = 67.871 and the upper bound is 69.5 + 1.629 = 71.129.

The confidence interval supports the conclusion of the hypothesis test because the advertised battery life of 72 hours falls within the confidence interval. This means that, based on the sample data, we can be 95% confident that the true population mean battery life is somewhere between 67.871 and 71.129 hours.