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 trials with a mean of and a standard deviation of 30 trials with a mean of 756.22 and a standard deviation of 117.63.

a) Find a ​98% confidence interval for the true speed of light from these statistics.

b) In words, what this interval means. 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 ​

c) What assumptions must you make in order to use your​method?

​a) A ​98% confidence interval for the true speed of light is (_____________kn/sec.,___________km/sec)​

(Round to two decimal places as​ needed.) ​b) In​ words, what does the ​98% confidence interval​ mean?

A. For all​ samples, ​98% of them will have a mean speed of light that falls within the confidence interval.

B. Any measurement of the speed of light will fall within this interval 98​% of the time.

C. With ​98% ​confidence, based on these​ data, the speed of light is between the lower and upper bounds of the confidence interval.

D. The confidence interval contains the true speed of light ​98% of the time. ​

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

A. The sample is drawn from a large population.

B. The data come from a distribution that is nearly normal.

C. The data came from a distribution that is nearly uniform.

D. The measurements are independent.

E. The measurements arise from a random sample or suitabily randomized experiment.

Answer 1

Step1: The 98% confidence interval estimate for the population mean of speed of light The summary statistics: The Sample Size is n = 30. The Sample Mean is ž = 756.22. The Sample Standard Deviation is s=117.63. Step2: The formula for calculating the degrees of freedom is df = n-1 = 30-1 =29 Therefore, the value of the degrees of freedom is df = 29 The level of significance is a =0.02. Step3: The table values are calculated by using the Excel2010 function = tinv(probability, degrees of freedom) tinv(0.02,29) = 2.462021 -12.462021 Therefore, the table value of the t-distribution is 10.01.29 2.462021 Step 4: The formula for calculating the 98% confidence interval for the population mean is X-tant en sus X + 2x2x-1 TH X-100212.30-1 sus X+0.0212.30-1 sus&+hool2 THE s S → 8-1022 In 117.63 <4756.22+(2.462021) =756.22-(2.462021) (117.63 30 30 117.63 117.63 =756.22-(2.462021) SuS756.22+(2.462021) 5.477226) 5.477226 =756.22-(2.462021) (21.4762) SUS756.22+(2.462021) (21.4762) =>756.22-52.87487<43756.22+52.87487 =>703.345154 3809.0949 7-tavna insusitarne -= 703.345154 3809.0949 (b) Step 5: Interpretation: We can be 98% confident that the true population mean of speed of light is between 703.3451 <4 <809.0949 (c) The assumptions of the t-distribution to construct confience interval for the population mean (4) The data come from a distribution is approximately normally distributed The sample of measurements is independent. The sample drawn from a population is randomized and it should be suitable random experiment.