please show all work thank you Problem 3 Thymus Weights 12 p

please show all work. thank you!

Problem 3. (Thymus Weights) [12 pts] As part of a study of the development of the thymus gland, researchers weighed the glands of five chick embryos after 14 days of incubation. The thymus weights (mg) were as follows: 29.6 21.5 28.0 34.6 44.9 State in words the population mean of this problem. Calculate the mean and standard deviation for this data. Construct a 90% confidence interval for the population mean. Interpret this confidence interval. (d) What assumptions are needed for the confidence interval constructed in part (c) to be valid?

Solution

a) y¯ = 31.72 is our best estimate for population mean.

b) Sample mean = 31.72

Sample standard deviation = 8.729

c) Find the value t: such that Pr {T t} = .05 for level 90%

Degree of freedom: df=5 1 = 4.

t-Table gives: t = 2.13 for 90% confidence

We had y¯ = 31.72, s = 8.73 then

SEy¯ = 8.73/ 5 = 3.90

Add cushion: t SEy¯ = 8.31 (90% confidence)

The interval is 31.72 ± 8.31 , i.e. (23.41, 40.03) for 90% confidence

d)All the values were independentlyand randomly sampled from a population .