Question

A research scholar wants to know how many times per hour a certain strand of virus reproduces. The mean is found to be 12.9 reproductions and the population standard deviation is known to be 2.2. If a sample of 358 was used for the study, construct the 90% confidence interval for the true mean number of reproductions per hour for the virus. Round your answers to one decimal place.

lower endpoint?

upper endpoint?

Answer 1

The 90% confidence interval for the true mean number of reproductions per hour for the virus is calculated to be (12.7, 13.1) when using a sample mean of 12.9, population standard deviation of 2.2, and sample size of 358.

Step-by-step explanation:

To construct the 90% confidence interval for the true mean number of reproductions per hour for a virus, we will use the formula for the confidence interval of the mean when the population standard deviation is known:

CI = μ ± Z*(σ/√n)

where μ is the sample mean, Z is the z-value from the standard normal distribution corresponding to the desired confidence level, σ is the population standard deviation, and n is the sample size.

Given that the mean (μ) is 12.9, the population standard deviation (σ) is 2.2, and the sample size (n) is 358, to find the Z value for a 90% confidence interval, we look up the z-value that corresponds to 90% in the middle of a standard normal distribution. This value is approximately 1.645.

The calculation of the margin of error (ME) is:

ME = Z*(σ/√n) = 1.645*(2.2/√358) ≈ 1.645*(0.1162) ≈ 0.1912

The confidence interval is thus:

Lower endpoint: 12.9 - 0.1912 = 12.7 (rounded to one decimal place)

Upper endpoint: 12.9 + 0.1912 = 13.1 (rounded to one decimal place)

Therefore, the 90% confidence interval for the true mean number of reproductions per hour for this virus is (12.7, 13.1).