The manufacturer of hardness testing equipment uses steel-ball indenters to penetrate metal that is being tested, however, the m

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The manufacturer of hardness testing equipment uses steel-ball indenters to penetrate metal that is being tested, however, the m

anufacturer 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?

Mathematics

1 answer:

Inessa [12.2K] · Answer 1 · 2026-04-02T01:42:00+03:00

Answer:

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

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

Based on the analysis, the 95% confidence interval is specified as (0.2789;3.055)

The question regarding the 95% confidence interval's ability to ascertain potential differences in measurements between the two indenters is as follows:

Indeed, the confidence interval does not include the value 0, thus indicating that the Diamond values significantly exceed those of the Steel Ball at a 5% significance level.

Step-by-step explanation:

Here is the dataset in consideration:

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

By calculating the differences between diamond and steel ball measurements, we create the dataset:

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

In the next step, we compute the mean difference

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

Following that, we determine the standard deviation of the differences, arriving at:

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

A confidence interval refers to "a range of values that’s likely to encompass a population value with a certain level of confidence. It is typically expressed as a percentage indicating where a population mean falls within an upper and lower limit."

The margin of error represents the extent to which values diverge above and below the sample statistic in a confidence interval.

Normal distribution, is described as a "probability distribution that is symmetric about the mean, illustrating that data points close to the mean occur more frequently than those further away."

The confidence interval for the mean is derived using the following formula:

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

To calculate the critical value $t_{\alpha/2}$ , we first determine the degrees of freedom using:

$df=n-1=9-1=8$

Given a confidence level of 0.95 or 95%, the appropriate critical value can be found using Excel, a calculator, or a table. The Excel command would be: "=-T.INV(0.025,9)". Therefore, we observe that $t_{\alpha/2}=2.31$ .

Finally, we can substitute all our findings into formula (1):

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

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

In this case, the 95% confidence interval is calculated as (0.2789;3.055)

In determining if the two indenters yield distinct measurements, we find that the confidence interval does not enclose zero, allowing us to conclude that Diamond readings greatly surpass Steel Ball readings at the 5% significance level.