Question

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.

1. 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.
2. 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:

1. Define the null and alternative hypothesis for this test in mathematical terms and in words.
2. Report the level of significance.
3. 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.)
4. Provide your conclusion and interpretation of the results. Should the null hypothesis be rejected? Why or why not?

Data for discussion talking points

1) Diameters data frame

    diameters
0        2.28
1        1.26
2        1.93
3        1.38
4        2.63
5        2.11
6        2.60
7        2.20
8        2.67
9        2.92
10       2.44
11       2.13
12       1.48
13       2.56
14       2.17
15       1.90
16       1.89
17       2.27
18       2.92
19       1.87
20       2.29
21       2.18
22       3.31
23       1.96
24       2.06
25       2.71
26       2.87
27       2.22
28       2.92
29       3.14
30       1.96
31       1.70
32       2.89
33       2.30
34       1.71
35       2.74
36       3.09
37       2.92
38       2.76
39       2.13
40       2.35
41       2.20
42       2.96
43       2.50
44       3.14
45       1.96
46       3.19
47       3.01
48       2.60
49       2.11

2) 90% confidence interval (unrounded) = (2.2734912846323323, 2.506108715367667)
90% confidence interval (rounded) = ( 2.27 , 2.51 )

99% confidence interval (unrounded) = (2.207661363228155, 2.5719386367718444)
99% confidence interval (rounded) = ( 2.21 , 2.57 )

3) z-test hypothesis test for population mean
test-statistic = 1.27
two tailed p-value = 0.2046

Answer 1

The hypothesis being tested is:

H₀: µ = 2.30 cm

H_a: µ > 2.30 cm

The level of significance is 0.01.

The test statistic is 1.27.

The p-value is 0.2046/2 = 0.1023.

Since the p-value (0.1023) is greater than the significance level (0.01), we cannot reject the null hypothesis.

Therefore, we cannot conclude that the average diameter of ball bearings from the manufacturing process is greater than 2.30 cm.