**Answer:**

So on this case the 95% confidence interval would be given by (0.2789;3.055)

**The 95% confidence interval to judge whether the two indeners result in different measurements is?**

Yes the confidence interval not contains the value 0 so we can conclude that the values for Diamond are significantly higher than the values for Steel Ball at 5% of significance.

**Step-by-step explanation:**

We have the following dataset:

specimen 1 2 3 4 5 6 7 8 9

Steel Ball 51 57 61 70 68 54 65 51 53

Diamond 53 55 63 74 69 56 68 51 56

If we calculate the differences diamond-steel ball we have this datase:

d: 2, -2, 2, 4, 1, 2, 3, 0, 3

The second step is calculate the mean difference

The third step would be calculate the standard deviation for the differences, and we got:

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".

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

(1)

In order to calculate the critical value we need to find first the degrees of freedom, given by:

Since the confidence is 0.95 or 95%, the value of and , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,9)".And we see that .

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

So on this case the 95% confidence interval would be given by (0.2789;3.055)

**The 95% confidence interval to judge whether the two indeners result in different measurements is?**

Yes the confidence interval not contains the value 0 so we can conclude that the values for Diamond are significantly higher than the values for Steel Ball at 5% of significance.