Question

A sports physician conducts an observational study to learn the average amount of time that 3,000 swimmers in the town can hold their breath underwater. He uses 150 sampling of 60 people. The average of the means of all the samplings is 72.7, and the standard deviation is 0.92. This is a histogram of the sampling distribution of the sample mean. Based on this data, with a 95% confidence interval the researchers can determine that the actual average amount of time the entire population can hold their breath under water is? Between and

Answer 1

95% confidence interval for the actual average amount of time the entire population can hold their breath under water is between 72.55 and 72.85.

Explanation:

Confidence Interval for the population mean is basically an interval of range of values where the true population mean can be found with a certain level of confidence.

Mathematically,

Confidence Interval = (Sample mean) ± (Margin of error)

Sample Mean = 72.7

Margin of Error is the width of the confidence interval about the mean.

It is given mathematically as,

Margin of Error = (Critical value) × (standard Error of the mean)

Critical value will be obtained using the t-distribution. This is because there is no information provided for the population mean and standard deviation.

To find the critical value from the t-tables, we first find the degree of freedom and the significance level.

Degree of freedom = df = n - 1 = 150 - 1 = 149

Significance level for 95% confidence interval

(100% - 95%)/2 = 2.5% = 0.025

t (0.025, 1499) = 1.975 (from the t-tables)

Standard error of the mean = σₓ = (σ/√n)

σ = standard deviation of the sample = 0.92

n = sampling size = 150

σₓ = (0.92/√150) = 0.0751176854 = 0.07512

95% Confidence Interval = (Sample mean) ± [(Critical value) × (standard Error of the mean)]

CI = 72.7 ± (1.975 × 0.07512)

CI = 72.7 ± 0.1483574288

95% CI = (72.55, 72.85)

95% Confidence interval = (72.55, 72.85)

Hope this Helps!!!