Question

The bloodhound is the mascot of John Jay College. Suppose we weigh n=8 randomly selected bloodhounds and get the following weights in pounds

85.6, 91.6, 105.9, 83.1, 102.1, 92.5, 108.8, 81.4

Assume bloodhound weight are normally distributed with unknown mean of μ pounds and an unknown standard deviation of σ pounds.

e) Suppose W has a t distribution with 7 degrees of freedom. If P(W > t) = .03 then what is t?  

f) Suppose W has a t distribution with 7 degrees of freedom. If P(W < t) = .03 then what is t?  

g) Calculate the 97th percentile of a standard normal distribution.  

h) Compute a 94% Confidence Interval for μ using your answers above.

i) Compute a 94% Prediction Interval for a single future bloodhound weight measurement using your answers above..

Answer 1

Answer:

e) P(W>t)=0.03, where W~X² with df 7

=> t = = Upper 97% point of chi-square distribution with df 7

=> t = 15.5

f) P(W<t) = 0.03, where W~X² with df 7

=> t = = Lower 3% point of chi-square distribution with df 7

=> t = 1.8

g) Suppose, x is the 97th percentile for standard normal distribution.

Thus, P(X<x)=0.97, where X~N(0,1)

=> x = = Upper 97% point of N(0,1) distribution

=> x = 1.881

h) In case population mean and population standard deviation are not known then sample mean and sample standard deviation are best points of estimation of population parameters (mean and standard deviation)

sample mean (m)= (85.6+91.6+105.9+83.1+102.1+92.5+108.8+81.4)/8= 93.87

Using above formula, sample standard deviation= 9.88

As sample size is very less (8 only), we will use t-statistic at 94% confidence interval.

(at 94% confidence level)= 1.77

upper confidence value= m+1.77*s= 93.87+1.77*3.49=100.05 pounds

Lower confidence value= m-1.77*s= 93.87-1.77*3.49.=87.68 pounds

Hence 94% confidence interval for population mean= (87.68 pounds,100.05 pounds)

i) In calculation of prediction interval, we will not use standard error instead we will be using sample standard deviation. Because Prediction interval is related to observation value while confidence interval is related to population mean. Other reason is that standard error represents variation in sampling mean distribution and sample standard deviation represents the variation in observations.

hence 94% prediction interval = (76.38 pounds, 111.35 pounds)