Final answer:
To construct the 85% confidence interval for virus reproductions per hour, the sample mean of 8.2, population standard deviation of 1.8, and sample size of 619 are used with a z-score of 1.036. The margin of error is calculated and added to and subtracted from the sample mean to get a range of 8.1 to 8.3 reproductions, rounded to one decimal place.
Step-by-step explanation:
To construct an 85% confidence interval for the true mean number of virus reproductions per hour, you can use the formula for the confidence interval based on the sample mean, the population standard deviation, and the sample size. Given that the sample mean is 8.2 reproductions, the population standard deviation is 1.8, and the sample size is 619, you would use the standard normal distribution to obtain the z-score that corresponds to the 85% confidence level.
The z-score for an 85% confidence level is approximately 1.036 because you are looking at the z-score that leaves 7.5% in each tail of the standard normal distribution (100% - 85% = 15%, and 15% / 2 = 7.5%). The formula for the confidence interval is:
Confidence Interval = Sample Mean ± (Z-score * (Population Standard Deviation / sqrt(Sample Size)))
Plugging in the values to the formula:
Confidence Interval = 8.2 ± (1.036 * (1.8 / sqrt(619)))
Calculating the margin of error:
Margin of Error = 1.036 * (1.8 / sqrt(619)) ≈ 0.075
Thus, the 85% confidence interval is:
Lower Limit = 8.2 - 0.075 ≈ 8.1
Upper Limit = 8.2 + 0.075 ≈ 8.3
So, the 85% confidence interval for the true mean number of reproductions per hour for the virus is from 8.1 to 8.3 reproductions, rounded to one decimal place.