A technician wants to estimate the average battery life of a new type of smart phone, so he tests 8 randomly selected phones & records the data below. Assuming battery life has a normal dist, make a 90% conf. int. for the mean battery life.

• A. I am 5% confident the average battery life is between (12.709, 13.191)
• B. I am 90% confident the average battery life is between (12.709, 13.191)
• C. I am 90% confident the average battery life is between (12.757, 13.143)
• D. I am 5% confident the average battery life is between (12.757, 13.143)

Answer: (C)

Verified step by step guidance

1. Step 1: Calculate the sample mean (\( \bar{x} \)) of the battery life data. Add all the battery life values together and divide by the number of observations (8). The formula is \( \bar{x} = \frac{\sum x_i}{n} \), where \( x_i \) are the individual data points and \( n \) is the sample size.
2. Step 2: Calculate the sample standard deviation (\( s \)) using the formula \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \). This measures the spread of the data around the mean.
3. Step 3: Determine the critical value (\( t^* \)) for a 90% confidence interval using the t-distribution table. Since the sample size is 8, the degrees of freedom (df) is \( n-1 = 7 \). Look up the critical value for a two-tailed test with 90% confidence level.
4. Step 4: Compute the margin of error (ME) using the formula \( ME = t^* \cdot \frac{s}{\sqrt{n}} \). This accounts for the variability in the sample mean.
5. Step 5: Construct the confidence interval using the formula \( \text{Confidence Interval} = \bar{x} \pm ME \). This provides the range within which the true mean battery life is likely to fall with 90% confidence.