A department store has determined that 25% of all their sales are credit sales. A random sample of 75 sales is selected and the proportion of credit sales in the sample is computed.

a) What is the probability that the same proportion will be greater than 0.34?

In a random sample of N, the most likely number of credit sales will be
m = 0.25 N,
and the standard deviation of the distribution is
sigma = sqrt (N*0.25*0.75) = 0.433sqrt N
When N= 75, m = 18.75 sigma = 3.75
If the number of credit sales is 0.34N, that is 25.5.
Using the normal distribution tool of
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html ,
I get the probability to be 3.6%

just to make sure i did this right.

What is the probability that the sample proportion will be between 0.196 and 0.354?

The answer i got is 84.12%

And the probability that the sample proportion will be less than 0.25 is 50%?

I have one question, from looking at the textbook, it says that the mean and standard deviation of a sample proportion is mu=p and sigma=sqrt(p(1-p)/n)

why would you not use this equation instead?

nevermind, it actually gave the same answer.

Well, it seems like you've got the hang of it! You calculated the probability of the sample proportion being greater than 0.34 correctly, and it's 3.6%. Good job!

As for the probability of the sample proportion being between 0.196 and 0.354, you calculated it to be 84.12%. That's quite high! It seems like there's a good chance the sample proportion falls within that range.

And yes, the probability of the sample proportion being less than 0.25 is indeed 50%. It's like flipping a coin - 50% chance it's heads, 50% chance it's tails. Simple as that!

Now, about your question regarding the mean and standard deviation of a sample proportion. You're right, the mean of a sample proportion is typically denoted as μ = p, where p is the population proportion. And the standard deviation is given by σ = sqrt(p(1-p)/n), where n is the sample size. It looks like that's the same equation you used to calculate the standard deviation earlier. So, you were on the right track!

Keep up the good work, and remember, statistics can be tricky, but with a little humor and a lot of practice, you'll conquer it!

It seems like you have correctly calculated the probabilities using the normal distribution.

Regarding your question about the mean and standard deviation of the sample proportion, the formula you mentioned is applicable when the sampling distribution is approximately normally distributed. In large samples, the sampling distribution of the sample proportion is indeed approximately normal, and the formula you mentioned can be used. However, for smaller sample sizes, the approximation may not hold, and a different formula, as mentioned earlier, can be used to estimate the mean and standard deviation. In this case, since you have a sample size of 75, it is appropriate to use the formula you used, which gave the same result.

It seems like you have correctly computed the probability in the first question, where the sample proportion is greater than 0.34.

To answer your second question, the formula you mentioned for the mean and standard deviation of a sample proportion, mu = p and sigma = sqrt(p(1-p)/n), is indeed another way to calculate the mean and standard deviation. In this case, p represents the proportion of interest (in this case, 0.25), and n represents the sample size (75 in this case). Using this formula should also yield the same result as the approach you used.

It's good that both formulas gave you the same answer, as it confirms the consistency of your calculations.