Normal Distribution Dog Weight

Question

Suppose the weights of full grown female beagles follow a normal distribution with a mean weight of $20$ pounds and a standard deviation of $3$ pounds.

1. A beagle was brought in for an exam and found to weigh $16.1$ pounds. Based on the provided weight model, approximately what percentage of beagles weigh less than $16.1$ pounds?

2. Based on the given distribution, the heaviest $15$% of full grown female beagles weigh at least how many pounds?

3. Let $Y=$ the number of full grown female beagles who weigh at least $20$ pounds in a random sample of $60$. What is the exact model of $Y$?

   a) Normal$(30, 3.873)$

   b) Binomial$(60, 0.20)$

   c) Binomial$(60, 0.50)$

   d) Normal$(20, 3)$

My answers:

1. $9.68%$%
2. $23.12$ lbs
3. I am stuck and need help. I think the answer is B but I am not sure. Any help?

Answer 1

You can get answers to (1) and (2) by standardizing and using printed normal tables. But using printed tables can require rounding, which may introduce slight errors.

Exact computations in R, where pnorm and qnorm are normal CDF and quantile functions, respectively, can give more exact answers. [The second parameter in these R procedures is the normal population standard deviation.] Thank you for showing your work; your answers are in good agreement with results from R.

pnorm(16.1, 20, 3)
[1] 0.09680048
qnorm(.85, 20, 3)
[1] 23.1093

Also, I agree with @patricio's comment.

Below is a graph of the density function of $\mathsf{Norm}(\mu=20, \sigma=3),$ with vertical lines (dotted red, and dashed blue) relevant to parts (1) and (2), respectively.

R code for figure:

curve(dnorm(x, 20, 3), 10, 30, lwd=2, ylab="PDF", 
      main = "Density of NORM920,3)")
 abline(h = 0, col="green2")
 abline(v = 16.1, col="red", lwd=2, lty="dotted")
 abline(v = 23.1, col="blue", lwd=2, lty="dashed")