Marla G. answered • 08/16/25
Masters Degree in Applied Statistics with 20+ Years of Work Experience
Hi Rebecca,
In case you don't have one of those fancy calculators, with all the stat. function buttons, what you really need is to understand the 'how', so here's what you need to do:
1) I like to start by writing down what I know, or am told in the problem.
a) For this problem, we're told that women's Ht is normally distributed with a mean=63.4 inches and a std. dev.=2.5 inches, or written another way: W.Ht. is N(63.4,2.5).
b) We've also been told that a branch of the military requires a person's ht be in the following interval:(58,80), and it also follows a ND(normal distribution).
2) The next step is to figure out how can we use what we know about the normal distribution to find the answer to the questions we've been given. The easiest way to move forward is to convert what we have from the ND(normal distribution) to what's called a (SND(standard normal distribution). Please recall an SND is just a ND with µ=1, and a σ=0.To convert a var with a ND to a var with a SND, use the following formula:
Z=(x-µ)/σ, where:
- Z is the standard normal variable,
- X is the original normal random variable
- μ is the mean of X
- σ is the standard deviation of X
I'll let you do the actually calculations, mainly because I'm not supposed to do your homework for you.
Finally, to answer the Part A, we need to solve the following probability: P(58 ≤ X ≤ 80). Again, we want convert these numbers to the SND using the formula I gave you above, this gives us
P(58-µ)/ο≤ X ≤ P(80-µ)/σP, substitute the values for the mean and std. dev. gives us
P{([58-63.4)/2.5] ≤ Z ≤ [(80-63.4)/2.5]}, this simplifies to
P (-2.16≤ Z ≤ 6.64)..Now you'll need to go to the tables that are likely in the back of your book and look up the 2 numbers, or 'Z-scores' you just calculated.
**Watchout: Not all tables are set up alike, so look for a small diagram somewhere on the page to see what side of the number the probability is for, or you can also think of it as "What part of the area under the normal curve" is the number the probability for.
MOVING ON TO PART B): Now we need to find the probability that a women's ht. falls in the top 1% or bottom 2%. We're going to do this one in a different, quicker way. You still need to calculate the Z-scores that correspond to the bottom 1%,(0.01) and top 2% , or 98% percentiles. 98 = (100-2):0.98), we do this because I want to be able to use the Z tables to find out what percentile corresponds to the standardized score that corresponds to the shortest 1% and the tallest 2% to answer the question. Once you have the 'Z' values calculated, there should be a table where you can look up the percentile of each number. **But, it's a little different than when you look for the number you calculate in the table, like you do for any hypothesis testing. This time, you want to look for the Z score that's 'closest' to the percentile of interest.In this case, look for the numbers 0.01 (lowest 1%), and 0.98( highest 2%). I would expect both numbers to be far out into the tails, because we know that for a SND about 97.5% of the data is within 3σ of the mean. You be more familiar with these facts: 68% of the data is within 1σ of the mean, and 95% of the data is within 2σ of the mean.
You may want to brush up on everything that knowing a variable is normally distributed tells you about the variable. You'll use that knowledge a lot in any stat class! Good Luck in your class, Marla
