Question

The manufacturer of hardness testing equipment uses steel-ball indenters to penetrate metal that is being tested, however, the manufacturer thinks it would be better to use a diamond indenter so that all ypes of metal can be tested. Because of differences between the two types of indenters, it is suspected that the two methods will produce different hardness readings. The metal speciments to be tested are large enough so that two indentions can be made. Therefore, the manufacturer uses both indenters on each specimen and compares the hardness readings. Construct a 95% confidence interval to judge whether the two indenters resul in different measurements. Note: A normal probability plot and boxplot indicate that the differences are approximately normally distributed with no outliers.Specimen Steel ball Diamond1 51 532 57 553 61 634 70 745 68 696 54 567 65 688 51 519 53 56Construct a 95% confidence interval to judge whether the two indenters result in different measurements, where the differences are computed as "diamond minus steel ball"The 95% confidence interval to judge whether the two indeners result in different measurements is?

Answer 1

Answer:

$1.667-2.31\frac{1.803}{\sqrt{9}}=0.2789$  

$1.667+2.31\frac{1.803}{\sqrt{9}}=3.055$  

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  

$\bar d= \frac{\sum_{i=1}^n d_i}{n}=1.667$

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

$s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =1.803$

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:  

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)  

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:  

$df=n-1=9-1=8$  

Since the confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$, 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 $t_{\alpha/2}=2.31$.

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

$1.667-2.31\frac{1.803}{\sqrt{9}}=0.2789$  

$1.667+2.31\frac{1.803}{\sqrt{9}}=3.055$  

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.