The mean weight of items coming from a production line is 83

The mean weight of items coming from a production line is 83 pounds. On the average 19 out of 20 of these individual weights are between 81 and 85 pounds. The average weights of six randomly selected items are plotted. What proportion of these points will fall between the limits 82 and 84 pounds? What is the precise nature of the random sampling hypothesis that you need to adopt to answer this question? Consider how it might be violated and what the effect of such a violation night be on your answer.

Solution

19 out of 20 that is 95% are between 81 and 85

About 95% of the values lie within 2 standard deviations of the mean

Mean =83

Therefore sd =1 ( 83-81)/2)

Standard error = 0.4082

Z value for 82, z=(82-83)/0.4082 =-2.45

Z value for 84, z=(84-83)/0.4082 =2.45

P( 82<x<84) = P( -2.45<z<2.45)

=P( z < 2.45) – P( z < -2.45)

0.9929 - 0.0071 = 0.9858

A rule that applies to normal distributions, stating that 68% of all data points fall within one standard deviation of the mean, 95% of all data points fall within two standard deviations of the mean, and 99.7% of all data points fall within three standard deviations of the mean.

If the sample is not random or the population is not normal, the result is not valid.