Question: IMPORTANT NOTE: Consider the following scenario based on the statistics and calculations provided further. Special Case # 1 The manufacturing process at a factory produces

IMPORTANT NOTE:

Consider the following scenario based on the statistics and calculations provided further.

Special Case # 1

"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".

Diameters data frame
Diameters
0 2.58
1 2.95
2 3.21
3 2.47
4 1.64
5 2.27
6 2.29
7 2.58
8 1.67
9 1.59
10 2.02
11 2.12
12 2.40
13 2.50
14 2.40
15 2.36
16 2.06
17 2.18
18 3.06
19 2.51
20 2.22
21 2.23
22 3.14
23 2.28
24 2.65
25 2.44
26 3.29
27 2.31
28 2.26
29 1.99
30 2.14
31 1.99
32 2.23
33 1.75
34 2.89
35 2.34
36 2.04
37 3.84
38 2.64
39 1.93
40 2.77
41 2.62
42 1.92
43 2.65
44 2.23
45 3.05
46 1.75
47 2.46
48 1.96
49 2.55

A. Confidence intervals:

90% confidence interval (unrounded) = (2.2720912846323325, 2.5047087153676673)

90% confidence interval (rounded) = ( 2.27 , 2.5 )

99% confidence interval (unrounded) = (2.2062613632281547, 2.570538636771845)

99% confidence interval (rounded) = ( 2.21 , 2.57 )

B. Hypothesis testing for the population mean:

z-test hypothesis test for population mean

test-statistic = 1.36

two tailed p-value = 0.1732

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.

Define the null and alternative hypothesis for this test in mathematical terms and in words.
Report the level of significance.
What valuable insights can you derive from the test statistic and the P-value? Keep in mind to 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? Explain!

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