Question

Dodge advertises on their website (http://www.dodge.com/en/2010/challenger/) that the
EPA estimated highway gas mileage of a 2010 Challenger is 25.0 miles per gallon (mpg).
Assuming the gas mileages have a standard deviation of 1.6 mpg, answer the following
questions.
a) What is the probability that the mean highway gas mileage of a random sample of 36
Challengers is between 24.5 mpg and 26.5 mpg?

Answer 1

Convert 24.5 and 26.5 to z scores. How many standard deviations are they from the mean?

Answer 2

24.5-25 / 1.6 = -.3125

Answer 3

26.5-25 / 1.6 = .9375

Answer 4

now i draw a blank :c

Answer 5

what is the exact formula i should be useing to solve?

Answer 6

after z = (x - μ) / σ

Answer 7

As Smoothmath's instruction, you should look up the table to find z score first, just like previous problem he has guided you!

Answer 8

z=.49,z= 1.53

Answer 9

Look like the next step is:
P ( z < 1.53 ) - P ( z < .49 )

Answer 10

1.04

Answer 11

I don't know much about stat, but the steps just like it!

Answer 12

I hate to say it, but I was in a rush and I actually gave pretty poor instruction on this one. Here's how to solve this problem:
Think about if I have a normally distributed data set, and I'm taking small samples from that data set. Maybe, for example, I keep taking samples of size 5 and looking at the MEAN for each sample. The means for those samples will all be AROUND the mean of the normal distribution that I'm choosing from. However, since I'm looking at pretty small samples, the deviation from that mean is pretty big.
Now, think about what happens when I increase the size of my samples. What happens, for example, if I look at samples of size 100 instead. My means for each sample will be much CLOSER to the mean of the distribution I'm choosing from.

To see this, think about coin flips. I should expect to flip about half heads. If I flip 10 coins at a time, it's not THAT unlikely that I flip 80% heads. However, if I'm looking at larger samples, for example 100 coins, then it becomes VERY unlikely that I flip 80% heads or anything else that far from the mean.

The Central Limit Theorem takes this idea and quantifies it. It says that if the standard deviation of the data set I'm choosing from is sigma and my sample size is n, then the new standard deviation will be
\[\sigma / \sqrt{n}\

Answer 13

So, for this problem, they give you:
Mean 25
Standard deviation 1.6
Sample size 36
So, for samples of that size, their mean is 25, and their standard deviation is 1.6/sqrt(36)

Answer 14

Use THAT number as the NEW standard deviation to calculate your z scores. Then use those z scores to find your probability.