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 25 trials with a mean of 756.22 and a standard deviation of 100.89. ​

(a) Find a 90​% confidence interval for the true speed of light from these statistics.
(​b) State 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?

Answer 1

a) The 90% confidence interval would be given by (721.716;790.724)

b) We are 90% confident that the true mean for the true speed of light is between (721.716;790.724)

c) We assume the following conditions:

1. Randomization
2. Independence
3. Deviation unknown
   `\sigma`

Explanation:

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

Part a

`\bar X=756.22` represent the sample mean

`\mu` population mean (variable of interest)

`s=100.89` represent the sample standard deviation

n=25 represent the sample size

90% confidence interval

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

The degrees of freedom are given by:

`df=n-1=25-1=24`

Since the Confidence is 0.90 or 90%, the value of
`\alpha=0.1` and
`\alpha/2 =0.05`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.05,24)".And we see that
`t_(\alpha/2)=1.71`

Now we have everything in order to replace into formula (1):

`756.22-1.71(100.89)/(√(25))=721.716`

`756.22+1.71(100.89)/(√(25))=790.724`

So on this case the 90% confidence interval would be given by (721.716;790.724)

Part b

We are 90% confident that the true mean for the true speed of light is between (721.716;790.724)

Part c

We assume the following conditions:

1. Randomization
2. Independence
3. Deviation unknown
   `\sigma`