1. ## Probability distribution question

I'm having really hard time with this question. Thanx in advance!

Assume that the daily distance travelled per car in Toronto has a normal distribution, with a mean of 4km(car-day) and a standard deviation of 1 km(car-day). In Toronto a random sample of 23 cars will be used and the sample mean will be calculated.
a) What probability model(distribution) describes the set of possible sample mean estimates
1) type of probability model(probability distribution)
2) its expected value
3) its standard deviation
b)
Would the probability model for the set of possible sample mean estimates change if the probability model for daily km's was a uniform distribution?
c)Give an answer to question a)1,2,3 if your sample(n=23) actually represented 40% of the entire population of interest. While the population retained the same expected value and standard deviation as the original population.
The city of Toronto contends that electric cars travel greater daily distances . To test this assertion they have taken a random sample of electric cars and measured their daily km. The results are shown below if you should need them

4.5 2.6 5.0 4.8 5.9 4.8 4.6 4.8 3.0 6.7 6.3 5.4 5.7 4.3 4.9 2.5 km (car-day)
sample mean = 4.73 km/day
sample standard deviation = 1.21 km(car-day)

and the second question is supposed to be a bit easier . I know that i'm supposed to find the z score and then convert it to find how many bags are underweight but i'm not sure about the other 3 parts . I don't know whether i should add or multiply the probabilities.

The weight of potato chips in a small-size bag is stated to be 5 ounces.The amount that the packing machine puts in these bags is believed to have a normal model with a mean of 5.1 ounces and a standard deviation of 0.08 ounces.

a) What fraction of all bags sold are underweight?
b) Some of the chips are sold in bargain packs of 5 bags, what is the probability that none of these 5 bags are underweight?
c) What is the probability that the mean weight of the 5 bags is below the stated amount?
d) What's the probability that the mean weight of a 20 bag case of potato chips is below 5 ounces?

2. Originally Posted by anna12345
I'm having really hard time with this question. Thanx in advance!

Assume that the daily distance travelled per car in Toronto has a normal distribution, with a mean of 4km(car-day) and a standard deviation of 1 km(car-day). In Toronto a random sample of 23 cars will be used and the sample mean will be calculated.
a) What probability model(distribution) describes the set of possible sample mean estimates
If the distribution of x is normal then the sample mean will also be normal.

It will have mean $\displaystyle \mu$ and st.dev $\displaystyle \frac{\sigma}{\sqrt{n}}$

Commit this to memory.

Originally Posted by anna12345
The weight of potato chips in a small-size bag is stated to be 5 ounces.The amount that the packing machine puts in these bags is believed to have a normal model with a mean of 5.1 ounces and a standard deviation of 0.08 ounces.

a) What fraction of all bags sold are underweight?
b) Some of the chips are sold in bargain packs of 5 bags, what is the probability that none of these 5 bags are underweight?
c) What is the probability that the mean weight of the 5 bags is below the stated amount?
d) What's the probability that the mean weight of a 20 bag case of potato chips is below 5 ounces?
Let's do these a couple at a time.

First you need to know that $\displaystyle Z = \frac{X-\mu}{\sigma}$

For a) You need to find $\displaystyle P(X<5) = P\left( Z<\frac{5-5.1}{0.08}\right) = \dots$

Using your answer in a) as $\displaystyle p$, b) is binomial where Y is the amount of bags so Y~Bi(n,p) = Y~Bi(5,p)

Now find $\displaystyle P(Y=0)$