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. a) Calculate the sample mean for this data. b) Calculate the sample variance for this data.. c) Calculate the sample standard deviation for this data. d) Calculate the maximurm likelihood estimate for using this data. 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( t)03 then what is t? g) Calculate the 97th percentile of a standard normal distribution. h) Compute a 94% Confidence Interval for using your answers above. i) Compute a 94% Prediction Interval for a single future bloodhound weight measurement using your answers above.. k) Copy your R script or other comments for the above into the text box here

All questions and round to 4 decimals points please. Thank you!

Answer 1

# data

> w = c(85.6, 91.6, 105.9, 83.1, 102.1, 92.5, 108.8, 81.4)

a)
> mean(w)
[1] 93.875

b)
> var(w)
[1] 111.7536

c)
> sd(w)
[1] 10.57136

d) MLE of $\sigma^2$ is $\dfrac{1}{n}\sum_{i=1}^n (X_i - \bar{X})^2$ . Using this, we can find MLE here:
> 1/8*sum((w-mean(w))^2)
[1] 97.78438

e)

> qt(0.03, df = 7, lower.tail = FALSE)
[1] 2.240879

f)
> qt(0.03, df = 7, lower.tail = TRUE)
[1] -2.240879

g)
> qnorm(0.97, lower.tail = TRUE)
[1] 1.880794

h) Confidence interval of $\mu$ is $(\bar{x}-t_{\alpha/2, n-1}s/\sqrt{n}, \bar{x}+t_{\alpha/2, n-1}s/\sqrt{n})$ . Using this formula, we can find the confidence interval:

> t_lower <- qt(0.03, df = 7, lower.tail = TRUE)
> t_upper <- qt(0.03, df = 7, lower.tail = FALSE)
> con_lower <- mean(w) + t_lower*sd(w)/sqrt(8)
> con_upper <- mean(w) + t_upper*sd(w)/sqrt(8)
> c(con_lower, con_upper)
[1] 85.49963 102.25037

Hence confidence interval is (85.49963, 102.25037)

i) Predictive interval of $\mu$ is $(\bar{x}-t_{\alpha/2, n-1}s\sqrt{1+1/n}, \bar{x}+t_{\alpha/2, n-1}s\sqrt{1+1/n})$ . Using this formula, we can find the predictive interval:

> pred_lower <- mean(w) + t_lower*sd(w)*sqrt(1+1/8)
> pred_upper <- mean(w) + t_upper*sd(w)*sqrt(1+1/8)
> c(pred_lower, pred_upper)
[1] 68.74888 119.00112

Hence predictive interval is (68.74888, 119.00112)