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.

Question

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.

esfloravaou · Accepted Answer

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=(n−1)s2χα22&amp;lt;σ2&amp;lt;(n−1)s2χ1−α22  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 · Accepted Answer

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 n−1=19, df=1.729  zα2=z0.052 - is the critical value at n−1=19, df=2.093  zα2=z0.012 - is the critical value at n−1=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=2668.15−45.44=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=2668.15−55=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=2668.15−75.18=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 · Accepted Answer

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 : χ1−α22=χ0.90, 192=11.6509 and  χα22=χ0.1, 192=27.2036 χ1−α22=χ0.95, 192=10.1170 and χα22=χ0.05, 19=30.1435 χ1−α22=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=(20−1)117.52227.2036&amp;lt;σ2&amp;lt;(20−1)117.52211.6509 =9646.327&amp;lt;σ2&amp;lt;22523.09 =98.21&amp;lt;σ&amp;lt;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=(20−1)117.52230.1435&amp;lt;σ2&amp;lt;(20−1)117.52210.1170 =8705.502&amp;lt;σ2&amp;lt;25937.95 =93.30&amp;lt;σ&amp;lt;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=(20−1)117.52236.1909&amp;lt;σ2&amp;lt;(20−1)117.5227.6327 =7250.85&amp;lt;σ2&amp;lt;34380.17 =85.15&amp;lt;σ&amp;lt;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

The manufacturer of LED lights would like to include as part

Answered question

Answer & Explanation

New Questions in College Statistics