The manufacturer of LED lights would like to include as part

Cheexorgeny

Cheexorgeny

Answered question

2021-12-19

The manufacturer of LED lights would like to include as part of their advertising the number of hours an LED is expected to work. A sample of 20 lights revealed the following number of hours worked.
26982628247428952672267524862852273326552765279424862627262427852692265425182650
Part 1: Draw normal probability plot for the above data and make comment.
Part 2: Assuming given data to be approximately normally distributed, Construct 90%, 95% and 99% confidence intervals for the mean number of hours that the LED lights of this brand has worked.
Part 3: Construct 90%, 95% and 99% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.

Answer & Explanation

esfloravaou

esfloravaou

Beginner2021-12-20Added 43 answers

Step 1
Introduction:
a) Confidence interval formula for the population mean is given by
C.I=x±zα2sdn
where x-bar is the sample mean
zα2 - is the critical value
sd - is the sample standard deviation
n - is sample size.
b) Confidence interval formula for the population standard deviation is given by
C.I=(n1)s2χα22<σ2<(n1)s2χ1α22
Here n is sample size
s2 - is the sample variance
χα22 - is the critical value
n - is the sample size
Step 2
Part 1: Normal probability plot

From the above normal probability plot we can observe that the given set of data follows normal distribution.Since, the all the data points are close to each other and to wards the straight line. Further, it confirms that the number of hours that the LED lights of this brand worked following normality.
Neil Dismukes

Neil Dismukes

Beginner2021-12-21Added 37 answers

Step 1
Part 2: 90%, 95% and 99% confidence intervals for the mean number of hours that the LED lights of this brand has worked.
For the given data we have,
x=2668.15, sd=117.52 and n=20
Critical value :
zα2=z0.12 - is the critical value at n1=19, df=1.729
zα2=z0.052 - is the critical value at n1=19, df=2.093
zα2=z0.012 - is the critical value at n1=19, df=2.861
90% confidence intervals for the mean number of hours that the LED lights of this brand has worked is
C.I=2668.15±1.729×117.5220
=2668.15±45.44
Lower limit=2668.1545.44=2622.71
Upper limit=2668.15+45.44=2713.59
So,we are 90% confident that the mean number of hours that the LED lights of this brand has worked is lies between 2622.71 and 2713.59
95% confidence intervals for the mean number of hours that the LED lights of this brand has worked is
C.I=2668.15±2.093×117.5220
=2668.15±55
Lower limit=2668.1555=2613.15
Upper limit=2668.15+55=2723.15
So,we are 95% confident that the mean number of hours that the LED lights of this brand has worked is lies between 2613.15 and 2713.15.
99% confidence intervals for the mean number of hours that the LED lights of this brand has worked is
C.I=2668.15±2.861×117.5220
=2668.15±75.18
Lower limit=2668.1575.18=2592.97
Upper limit=2668.15+75.18=2743.33
So,we are 99% confident that the mean number of hours that the LED lights of this brand has worked is lies between 2592.97 and 2743.33.
nick1337

nick1337

Expert2021-12-28Added 777 answers

Step 1
Part 3: 90%, 95% and 99% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.
For the given data we have,
x¯=266.15

sd=117.52
n=20
Critical value :
χ1α22=χ0.90, 192=11.6509 and 
χα22=χ0.1, 192=27.2036
χ1α22=χ0.95, 192=10.1170 and
χα22=χ0.05, 19=30.1435
χ1α22=χ0.99, 192=7.6327 and
χα22=χ0.01, 192=36.1909
90% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.
C.I=(201)117.52227.2036<σ2<(201)117.52211.6509
=9646.327<σ2<22523.09
=98.21<σ<150.077
So,we are 90% confident that the Standard deviation of hours that the LED lights of this brand has worked is lies between 98.21 and 150.077
95% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.
C.I=(201)117.52230.1435<σ2<(201)117.52210.1170
=8705.502<σ2<25937.95
=93.30<σ<161.05
So,we are 95% confident that the Standard deviation of hours that the LED lights of this brand has worked is lies between 93.30 and 161.05
99% confidence intervals for the standard deviation of the number of hours that the LED lights of this brand worked.
C.I=(201)117.52236.1909<σ2<(201)117.5227.6327
=7250.85<σ2<34380.17
=85.15<σ<185.41
So,we are 99% confident that the Standard deviation of hours that the LED lights of this brand has worked is lies between 85.15and 185.41

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