Question

A scientist measured the speed of light. His values are in​ km/sec and have​ 299,000 subtracted from them. He reported the results of 21 trials with a mean of 756.22 and a standard deviation of 109.59.

​1) Find a 95​% confidence interval for the true speed of light from these statistics. Round to two decimals as needed.

​2) In words, what does the 95% confidence interval mean. Keep in mind that the speed of light is a physical constant​ that, as far as we​ know, has a value that is true throughout the universe.

a) Any measurement of the speed of light will fall within this interval 95% of the time.

b) The confidence interval contains the true speed of light 95% of the time.

c) With 95% confidence, based on the data, the speed of light is between the lower and upper bounds of the confidence interval.

d) For all samples, 95% of them will have a mean speed of light that falls within the confidence interval.

​3) What assumptions must you make in order to use your​ method? Select all that apply:

- The measurements are independent

- The data comes from a distribution that is nearly normal

- The measurements arise from a random sample or suitably randomized experiment

- The data comes from a distribution that is nearly uniform.

- The sample is drawn from a large population

Answer 1

sample mean, xbar = 756.22
sample standard deviation, s = 109.59
sample size, n = 21
degrees of freedom, df = n - 1 = 20

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, tc = t(α/2, df) = 2.086

CI = (xbar - tc * s/sqrt(n) , xbar + tc * s/sqrt(n))
CI = (756.22 - 2.086 * 109.59/sqrt(21) , 756.22 + 2.086 * 109.59/sqrt(21))
CI = (706.33 , 806.11)

2)
With 95% confidence, based on the data, the speed of light is between the lower and upper bounds of the confidence interval.

3)
- The measurements are independent

- The data comes from a distribution that is nearly normal

- The measurements arise from a random sample or suitably randomized experiment