Question

2.   A rivet is to be inserted into a hole. If the standard deviation of hole diameter exceeds 0.02 mm, there is an unacceptably high probability that the rivet will not fit. A random sample of n = 15 parts is selected, and the hole diameter is measured. The sample standard deviation of the hole diameter measurements is s = 0.016mm. At α = 0.05 conduct a hypothesis test to investigate to indicate that the standard deviation of hole diameter exceeds 0.02 mm. To gain full credit, you should provide the following 1-8:
1.   State and check the modeling assumptions.
2.   Define the parameter of interest.
3.   State the hypotheses.
4.   Calculate the value of the test statistic. What is the distribution of the test statistic?
5.   Find the p-value using the appropriate table.
6.   State the decision and the conclusion in the context of the problem.
7.   Calculate a 95% confidence for σ and interpret your interval in the context of this problem.
8.   Use the confidence bound in part 7 to test the hypothesis.

Answer 1

2.

Given that,
population standard deviation (σ)=0.02
sample standard deviation (s) =0.016
sample size (n) = 15
we calculate,
population variance (σ^2) =0.0004
sample variance (s^2)=0.000256
null, Ho: σ =0.02
alternate, H1 : σ >0.02
level of significance, α = 0.05
from standard normal table,right tailed ᴪ^2 α/2 =23.685
since our test is right-tailed
we use test statistic chisquare ᴪ^2 =(n-1)*s^2/o^2
ᴪ^2 cal=(15 - 1 ) * 0.000256 / 0.0004 = 14*0.000256/0.0004 = 8.96
| ᴪ^2 cal | =8.96
critical value
the value of |ᴪ^2 α| at los 0.05 with d.f (n-1)=14 is 23.685
we got | ᴪ^2| =8.96 & | ᴪ^2 α | =23.685
make decision
hence value of | ᴪ^2 cal | < | ᴪ^2 α | and here we do not reject Ho
ᴪ^2 p_value =0.8336
ANSWERS
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1.
Depending on the statistical analysis, the assumptions may differ.
A few of the most common assumptions in statistics are normality, linearity, and equality of variance.
Normality assumes that the continuous variables to be used in the analysis are normally distributed
2.
A parameter of interest is what your data is focused on.
When you don't have data about the entire population from a census,
you will often use a sample statistic to approximate the parameter of interest.
Parameter means some kind of number, like an average or a median. here, standard deviation
3.
A hypothesis states your predictions about what your research will find.
It is a tentative answer to your research question that has not yet been tested. For some research projects,
you might have to write several hypotheses that address different aspects of your research question
4.
null, Ho: σ =0.02
alternate, H1 : σ >0.02
test statistic: 8.96
5.
critical value: 23.685
p-value:0.8336
6.
decision: do not reject Ho
we do not have enough evidence to support the claim that If the standard deviation of hole diameter exceeds 0.02 mm,

7.
CONFIDENCE INTERVAL FOR STANDARD DEVIATION
ci = (n-1) s^2 / ᴪ^2 right < σ^2 < (n-1) s^2 / ᴪ^2 left
where,
s = standard deviation
ᴪ^2 right = (1 - confidence level)/2
ᴪ^2 left = 1 - ᴪ^2 right
n = sample size
since alpha =0.05
ᴪ^2 right = (1 - confidence level)/2 = (1 - 0.95)/2 = 0.05/2 = 0.025
ᴪ^2 left = 1 - ᴪ^2 right = 1 - 0.025 = 0.975
the two critical values ᴪ^2 left, ᴪ^2 right at 14 df are 26.1189 , 5.629
s.d( s )=0.02
sample size(n)=15
confidence interval for σ^2= [ 14 * 0.0004/26.1189 < σ^2 < 14 * 0.0004/5.629 ]
= [ 0.0056/26.1189 < σ^2 < 0.0056/5.6287 ]
[ 0.0002 < σ^2 < 0.001 ]
and confidence interval for σ = sqrt(lower) < σ < sqrt(upper)
= [ sqrt (0.0002) < σ < sqrt(0.001), ]
= [ 0.0146 < σ < 0.0315 ]
8.
95% confidence for σ and interpret your interval [ 0.0146 < σ < 0.0315 ]