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 perform the test to determine if the average battery life differs from the advertised life of 72 hours, we will use the test statistic method.
Step 1: State the null and alternative hypotheses:
- Null hypothesis (H0): The average battery life equals 72 hours (μ = 72).
- Alternative hypothesis (Ha): The average battery life differs from 72 hours (μ ≠ 72).
Step 2: Choose the significance level (alpha):
- The significance level (α) is given as 0.05.
Step 3: Identify the test statistic and the corresponding probability distribution:
- Since we are comparing a sample mean to a population mean with known standard deviation, we will use the Z-test.
- The test statistic formula is: Z = (x̄ - μ) / (σ / √n), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.
Step 4: Compute the test statistic:
- Given data:
- Sample mean (x̄) = 69.5 hours
- Population mean (μ) = 72 hours (from the advertisement)
- Population standard deviation (σ) = 5 hours
- Sample size (n) = 36
- Calculating the test statistic:
Z = (69.5 - 72) / (5 / √36) = -2.5 / (5 / 6) = -2.5 * (6 / 5) = -3
Step 5: Determine the rejection region:
- Since it is a two-tailed test and the significance level (α) is 0.05, we have to split the alpha value into two equal parts: 0.05/2 = 0.025.
- Using a Z-table or standard normal distribution calculator, we find that the critical Z-values for a two-tailed test at α = 0.025 are approximately -1.96 and 1.96.
Step 6: Make a decision:
- Since the calculated test statistic (-3) falls outside the rejection region (-1.96 to 1.96), we can reject the null hypothesis.
- Thus, we have evidence to conclude that the average battery life differs from the advertised life of 72 hours.
b. To compute the 95% confidence interval estimate for the mean battery life, we will use the formula:
- Confidence interval = x̄ ± (Z * (σ / √n)), where x̄ is the sample mean, Z is the critical value for the desired level of confidence, σ is the population standard deviation, and n is the sample size.
Using the same values from the given data:
- Sample mean (x̄) = 69.5 hours
- Population standard deviation (σ) = 5 hours
- Sample size (n) = 36
- From the Z-table or standard normal distribution calculator, the critical Z-value for a 95% confidence level is approximately 1.96.
Calculating the confidence interval:
Confidence interval = 69.5 ± (1.96 * (5 / √36))
Confidence interval = 69.5 ± (1.96 * (5 / 6))
Confidence interval = 69.5 ± 1.96(0.833)
Confidence interval = 69.5 ± 1.63268
Confidence interval = [67.86732, 71.13268]
The confidence interval suggests that we are 95% confident that the true mean battery life falls within the range of 67.86732 hours to 71.13268 hours. Since the hypothesized value of 72 hours falls outside this interval, it supports the conclusion from the hypothesis test that the average battery life differs from the advertised life.