Question

In this discussion, you will apply the statistical concepts and techniques covered in this week's reading to calculate a confidence interval and perform hypothesis testing for a manufacturing process. The manufacturing process at a factory produces ball bearings that are sold to automotive manufacturers. The factory wants to estimate the average diameter of a ball bearing that is in demand to ensure that it is manufactured within the specifications. Suppose they plan to collect a sample of 50 ball bearings and measure their diameters to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. The sample of size 50 was generated using Python's numpy module. This data set will be unique to you, and therefore your answers will be unique as well. Run Step 1 in the Python script to generate your unique sample data. Check to make sure your sample data is shown in your attachment. In your initial post, address the following items. Be sure to answer the questions about both confidence intervals and hypothesis testing. In the Python script, you calculated the sample data to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. These confidence intervals were created using the Normal distribution based on the assumption that the population standard deviation is known and the sample size is sufficiently large. Report these confidence intervals rounded to two decimal places. See Step 2 in the Python script. Interpret both confidence intervals. Make sure to be detailed and precise in your interpretation. It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm? Perform a hypothesis test for the population mean at alpha = 0.01. In your initial post, address the following items: Define the null and alternative hypothesis for this test in mathematical terms and in words. Report the level of significance. Include the test statistic and the P-value. See Step 3 in the Python script. (Note that Python methods return two-tailed P-values. You must report the correct P-value based on the alternative hypothesis.) Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not? The Data: Diameters data frame diameters 0 3.59 1 1.80 2 2.62 3 2.54 4 2.48 5 2.67 6 2.90 7 3.08 8 2.57 9 2.43 10 2.31 11 2.49 12 2.49 13 2.02 14 3.45 15 2.47 16 2.00 17 2.72 18 2.71 19 2.60 20 2.21 21 3.17 22 2.74 23 2.89 24 2.82 25 3.26 26 2.51 27 2.85 28 2.75 29 2.23 30 2.67 31 3.25 32 2.99 33 2.63 34 2.38 35 2.51 36 2.45 37 3.01 38 3.23 39 2.41 40 2.92 41 4.00 42 1.96 43 2.22 44 3.33 45 2.81 46 2.34 47 2.73 48 2.29 49 2.87 90% confidence interval (unrounded) = (2.571091284632333, 2.8037087153676676) 90% confidence interval (rounded) = (2.57, 2.8) 99% confidence interval (unrounded) = (2.505261363228155, 2.8695386367718454) 99% confidence interval (rounded) = (2.51, 2.87) z-test hypothesis test for population mean test-statistic = 6.33 two-tailed p-value = 0.0

Name: in this discussion you will apply the statistical concepts and techniques covered in this weeks reading to calculate a confidence interval and perform hypothesis testing for a manufacturing 88581
Uploaded: 2022-01-23T05:18:11-08:00
Duration: 11 min 8 s
Channel: Umar Sohail Qureshi
Description: in this discussion you will apply the statistical concepts and techniques covered in this weeks reading to calculate a confidence interval and perform hypothesis testing for a manufacturing 88581

The manufacturing process at a factory produces ball bearings that are sold to automotive manufacturers. The factory wants to estimate the average diameter of a ball bearing that is in demand to ensure that it is manufactured within the specifications.

Suppose they plan to collect a sample of 50 ball bearings and measure their diameters to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process.

The sample of size 50 was generated using Python's numpy module. This data set will be unique to you, and therefore your answers will be unique as well. Run Step 1 in the Python script to generate your unique sample data. Check to make sure your sample data is shown in your attachment.

In your initial post, address the following items. Be sure to answer the questions about both confidence intervals and hypothesis testing.

In the Python script, you calculated the sample data to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. These confidence intervals were created using the Normal distribution based on the assumption that the population standard deviation is known and the sample size is sufficiently large. Report these confidence intervals rounded to two decimal places. See Step 2 in the Python script.

Interpret both confidence intervals. Make sure to be detailed and precise in your interpretation.

It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm?

Perform a hypothesis test for the population mean at alpha = 0.01.

In your initial post, address the following items:

Define the null and alternative hypothesis for this test in mathematical terms and in words.

Report the level of significance.

Include the test statistic and the P-value. See Step 3 in the Python script. (Note that Python methods return two-tailed P-values. You must report the correct P-value based on the alternative hypothesis.)

Provide your conclusion and interpretation of the results.

Should the null hypothesis be rejected? Why or why not?

The Data:

Diameters data frame
diameters
0 3.59
1 1.80
2 2.62
3 2.54
4 2.48
5 2.67
6 2.90
7 3.08
8 2.57
9 2.43
10 2.31
11 2.49
12 2.49
13 2.02
14 3.45
15 2.47
16 2.00
17 2.72
18 2.71
19 2.60
20 2.21
21 3.17
22 2.74
23 2.89
24 2.82
25 3.26
26 2.51
27 2.85
28 2.75
29 2.23
30 2.67
31 3.25
32 2.99
33 2.63
34 2.38
35 2.51
36 2.45
37 3.01
38 3.23
39 2.41
40 2.92
41 4.00
42 1.96
43 2.22
44 3.33
45 2.81
46 2.34
47 2.73
48 2.29
49 2.87

90% confidence interval (unrounded) = (2.571091284632333, 2.8037087153676676)
90% confidence interval (rounded) = (2.57, 2.8)

99% confidence interval (unrounded) = (2.505261363228155, 2.8695386367718454)
99% confidence interval (rounded) = (2.51, 2.87)

z-test hypothesis test for population mean
test-statistic = 6.33
two-tailed p-value = 0.0

Added by Robert P.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Umar Sohail Qureshi

01/23/2022

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In this discussion, you will apply the statistical concepts and techniques covered in this week's reading to calculate a confidence interval and perform hypothesis testing for a manufacturing process. The manufacturing process at a factory produces ball bearings that are sold to automotive manufacturers. The factory wants to estimate the average diameter of a ball bearing that is in demand to ensure that it is manufactured within the specifications. Suppose they plan to collect a sample of 50 ball bearings and measure their diameters to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. The sample of size 50 was generated using Python's numpy module. This data set will be unique to you, and therefore your answers will be unique as well. Run Step 1 in the Python script to generate your unique sample data. Check to make sure your sample data is shown in your attachment. In your initial post, address the following items. Be sure to answer the questions about both confidence intervals and hypothesis testing. In the Python script, you calculated the sample data to construct a 90% and 99% confidence interval for the average diameter of ball bearings produced from this manufacturing process. These confidence intervals were created using the Normal distribution based on the assumption that the population standard deviation is known and the sample size is sufficiently large. Report these confidence intervals rounded to two decimal places. See Step 2 in the Python script. Interpret both confidence intervals. Make sure to be detailed and precise in your interpretation. It has been claimed from previous studies that the average diameter of ball bearings from this manufacturing process is 2.30 cm. Based on the sample of 50 that you collected, is there evidence to suggest that the average diameter is greater than 2.30 cm? Perform a hypothesis test for the population mean at alpha = 0.01. In your initial post, address the following items: Define the null and alternative hypothesis for this test in mathematical terms and in words. Report the level of significance. Include the test statistic and the P-value. See Step 3 in the Python script. (Note that Python methods return two-tailed P-values. You must report the correct P-value based on the alternative hypothesis.) Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not? The Data: Diameters data frame diameters 0 3.59 1 1.80 2 2.62 3 2.54 4 2.48 5 2.67 6 2.90 7 3.08 8 2.57 9 2.43 10 2.31 11 2.49 12 2.49 13 2.02 14 3.45 15 2.47 16 2.00 17 2.72 18 2.71 19 2.60 20 2.21 21 3.17 22 2.74 23 2.89 24 2.82 25 3.26 26 2.51 27 2.85 28 2.75 29 2.23 30 2.67 31 3.25 32 2.99 33 2.63 34 2.38 35 2.51 36 2.45 37 3.01 38 3.23 39 2.41 40 2.92 41 4.00 42 1.96 43 2.22 44 3.33 45 2.81 46 2.34 47 2.73 48 2.29 49 2.87 90% confidence interval (unrounded) = (2.571091284632333, 2.8037087153676676) 90% confidence interval (rounded) = (2.57, 2.8) 99% confidence interval (unrounded) = (2.505261363228155, 2.8695386367718454) 99% confidence interval (rounded) = (2.51, 2.87) z-test hypothesis test for population mean test-statistic = 6.33 two-tailed p-value = 0.0

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Transcript

00:01 In this problem, what we want to do is sample 100 students from several schools in a school system to estimate the mean weight of sixth graders in that school system.

00:12 So one way that we might do this is by dividing the school system up into strata, which are non -overlapping groups, so a reasonable way to do this might be dividing them up by school.

00:23 So what we're given is some information about how many individuals or how many sixth grade students are at each school, as well as a sample standard deviation from some previous data.

00:35 So the question might be, if we're limited to only choosing 100 students, how do we go about sampling, or how many do we know to sample, how do we know how many to sample from each of these three schools, to have everyone proportionally represented? so if we have information about the sample standard deviations, one way that we might do that to calculate the sample size for each school is by multiplying that school or that stratum's number of individuals times the standard deviation of observations for that school or stratum and then dividing by the sum of each stratum size times the standard deviation.

01:28 So in the denominator here we would have number of people in the first school times the sample standard deviation from that school plus number of people in the second school times that standard deviation plus number of people in the third school times the third standard deviation and again this will only work if we have that sample data indicating what the standard deviation would be and then what we do is we take this proportion and we multiply that by m which is the total that we want in the sample.

02:02 So this will give us a proportion of individuals that we should choose, and then we scale that up by how many people we would like to have.

02:09 So let's go ahead and calculate the sample size of students that we need from school a based on these data.

02:21 So for school a, what i'm going to do is take number of people at school a times standard deviation from school a.

02:36 Divide by the sum of those quantities mentioned before.

02:46 Then we multiply that by the sample size that we want, which here is 100.

02:52 Okay, let's fill in some of this information.

02:55 So in the numerator, the number of people at school a is 310, or number of sixth grade students.

03:03 So we'll have 310 times the standard deviation, which is 3.

03:11 This 100 can be multiplied in the numerator, excuse me and then in the denominator what we're going to have is 310 times 3 n2 times s 2 so that's number of people at school 2 times that standard deviation so we'll have times 420 times 12 plus 516 times 6 then simplifying that our numerator will become 93 ,000 and our denominator will become 9 ,066.

03:58 And then if you divide those, what you should get is 10 .2581.

04:06 Now remember that this is representing a sample size.

04:09 That's representing a number of students.

04:11 And we can't choose fractional numbers or decimal numbers of students.

04:15 So what we need to do is round this to the nearest whole number, which would give us 10 students.

04:21 We would round this down.

04:29 Okay, let's follow a similar process for school b.

04:41 We're going to calculate number of students from school b.

04:44 It's going to be n2 times s2.

04:51 And notice that our denominator is not going to change in this formula.

04:56 So we could just take the quantity we already computed for it, not reinvent the wheel here.

05:02 Save ourselves a little bit of work.