Answer:
So on this case the 95% confidence interval would be given by (-1.152;5.152).
Should we conclude that the average time it takes experts to crack the safes does not differ by model at the 5% significance level?
A) Yes, the 95% confidence interval for ?D contains 0.
Step-by-step explanation:
Previous concepts
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".
Solution
Let put some notation
x=value for Safe 1 , y = value for Safe 2
x: 103, 90, 64, 120, 104, 92, 145, 106, 76
y: 101, 94, 58, 112, 103, 90, 140, 110, 74
The first step is calculate the difference and we obtain this:
d: 2, -4, 6, 8, 1, 2, 5, -4, 2
The second step is calculate the mean difference
The third step would be calculate the standard deviation for the differences, and we got:
The next step is calculate the degrees of freedom given by:
Now we need to calculate the critical value on the t distribution with 8 degrees of freedom. The value of and
, so we need a quantile that accumulates on each tail of the t distribution 0.025 of the area.
We can use the following excel code to find it:"=T.INV(0.025;8)" or "=T.INV(1-0.025;8)". And we got
The confidence interval for the mean is given by the following formula:
(1)
Now we have everything in order to replace into formula (1):
So on this case the 95% confidence interval would be given by (-1.152;5.152).
Should we conclude that the average time it takes experts to crack the safes does not differ by model at the 5% significance level?
A) Yes, the 95% confidence interval for D contains 0.