Question

A dishwasher has a mean lifetime of 11 years with an estimated standard deviation of 2.07 years. The lifetime of a dishwasher is normally distributed. Suppose you are a manufacturer and you take a sample of 10 dishwashers that you made. Find the probability that the sample would have a mean lifetime less than 10.49 years? Round to four decimal places.

Answer 1

To find the probability that the sample mean lifetime of the dishwashers is less than 10.49 years, calculate the standard error and z-score. Then, use a standard normal distribution table to find the probability.

Step-by-step explanation:

To find the probability that the sample mean lifetime of the dishwashers is less than 10.49 years, we can calculate the z-score and use a standard normal distribution table.

First, we calculate the standard error of the sample mean using the formula:

Standard Error = Standard Deviation / sqrt(n)

Plugging in the values, we get:

Standard Error = 2.07 / sqrt(10) = 0.654

Next, we calculate the z-score:

Z = (Sample Mean - Population Mean) / Standard Error

Plugging in the values, we get:

Z = (10.49 - 11) / 0.654 = -0.508

Now, we can look up the z-score in the standard normal distribution table to find the probability of getting a z-score less than -0.508. The probability is approximately 0.3057.

Therefore, the probability that the sample would have a mean lifetime less than 10.49 years is approximately 0.3057.

Answer 2

The probability that the sample would have a mean lifetime less than 10.49 years is approximately 0.4313.

Step-by-step explanation:

To find the probability that the sample would have a mean lifetime less than 10.49 years, we need to standardize the sample mean using the standard deviation of the population. We can use the formula for the standard error of the mean:

Standard Error of the Mean (SE) = Standard Deviation (σ) / Square Root of Sample Size (n)

Plugging in the values for the standard deviation (2.07 years) and the sample size (10), we can calculate the standard error of the mean:

SE = 2.07 / sqrt(10) ≈ 0.6542 years

Next, we need to calculate the z-score, which measures how many standard errors the sample mean is from the population mean. The formula for the z-score is:

z = (Sample Mean - Population Mean) / Standard Error of the Mean

Plugging in the values for the sample mean (10.49 years), the population mean (11 years), and the standard error of the mean (0.6542 years), we can calculate the z-score:

z = (10.49 - 11) / 0.6542 ≈ -0.1728

Finally, we can use a standard normal distribution table or a calculator to find the probability that the z-score is less than -0.1728. Using either method, we find that the probability is approximately 0.4313. Therefore, the probability that the sample would have a mean lifetime less than 10.49 years is approximately 0.4313.