MATH1127: Introduction to Statistics (60022)
8.1 A Single Population Mean using the Normal Distribution
Unit 3 Chapter
An incubation period is a time between when you contract a virus and when your symptoms start. By surveying randomly selected local hospitals, a researcher was able to obtain a sample of incubation periods of 45 patients, and the sample average was found to be 6.2 days. Assume that the population standard deviation is 1.7 days. Construct $\overline{6} 94 \%$ confidence interval for the average incubation period of the novel coronavirus.
1. Procedure: Select an answer
ii. Assumptions: (select everything that applies)
Normal population
Sample size is greater than 30
Simple random sample
Population standard deviation is known
Population standard deviation is unknown
The number of positive and negative responses are both greater than 10
iii. Unknown parameter: Select an answer
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Step 1 ：The problem is asking for a 94% confidence interval for the average incubation period of the novel coronavirus. The sample size is 45, the sample mean is 6.2 days, and the population standard deviation is 1.7 days.

Step 2 ：The formula for a confidence interval is: \(\bar{x} \pm Z \frac{\sigma}{\sqrt{n}}\) where \(\bar{x}\) is the sample mean, \(Z\) is the Z-score corresponding to the desired level of confidence, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step 3 ：The Z-score for a 94% confidence interval can be found using a Z-table or a calculator that can compute probabilities for the standard normal distribution. The Z-score that corresponds to a 94% confidence interval is approximately 1.88.

Step 4 ：Substitute the given values into the formula: sample_mean = 6.2, population_std_dev = 1.7, sample_size = 45, z_score = 1.88.

Step 5 ：Calculate the standard error: standard_error = 0.2534210374499761

Step 6 ：Calculate the margin of error: margin_of_error = 0.4764315504059551

Step 7 ：Calculate the confidence interval: confidence_interval = (5.723568449594045, 6.6764315504059555)

Step 8 ：The 94% confidence interval for the average incubation period of the novel coronavirus is approximately \(\boxed{(5.72, 6.68)}\) days.