## Question 500:

1

a)      Find the probability that a student had a score higher than 325?

a.       Find the z-score for the score of 325 by subtracting the mean from this and dividing the result by the standard deviation:

b.      (325-276.1)/34.4 = a z-score of 1.42. Use the percentile to z-score calculator to find the area under the normal curve above 1.42, you should get 7.78% probability of getting a score higher than 325

b)      Find the probability that a student had a score between 250 and 305?

a.       For between area questions we need to find the larger then subtract the smaller area.

b.      1st z-score is (305-276.1)/34.4 = .84 and the 2nd z-score is (250-276.1)/34.4 = -.758 (notice the negative sign).

c.       Looking up the area for both using the z-score to percentile calculator gets us areas of 79.95% and 22.42%

d.      Subtracting the two gets us an area of 79.95-22.42 = 57.53 meaning that 57.53% probability a students score is between 250 and 305

c)      What percentage of the students had a test score that is greater than 250?

a.       The z-score is (250-276.1)/34.4 = -.758 and the area above this is 77.57%

d)      If 2000 students are randomly selected, how many will have a test score that is less than 300?

a.       The z-score is (300-276.1)/34.4 = .6947 and has the area of 75.64% below a score of 300. If 2000 are selected, we'd expect .7564*2000 = 1512.8 of them to have scores below 300.

e)      What is the lowest score that would place a student in the top 5% of the scores?

a.       Construct an equation and solve for the unknown score

b.      First find the z-score for the area above 5% by entering .95 and in the percentile to z-score calculator and you should get 1.642. Now use this z-score to solve for

c.       ( x -276.1 )/34.4 = 1.642

d.      x-276.1 = 56.48

e.       x = 332.58 or you'd need a score of at least 332.58 or rounding up to 333 to be in the top 5% of scorers.

f)        What is the highest score that would place a student in the bottom 25% of the scores?

a.       Same process as in e, the z-score for the bottom 25% is -.6742.

b.      ( x -276.1 )/34.4 = -.6742

c.       x-276.1 = -23.19

d.      x = 252.91

e.       So a score of 252.91 is the highest a student can score to still be scoring in the lowest 25%.

g)      A random sample of 60 students is drawn from this population. What is the probability that the mean test score is greater than 300?

a.       We would use the z-score for a sample mean formula here. The only difference is that we use the standard error of the mean (SEM) instead of the standard deviation to calculate z. The SEM is the population standard deviation divided by the square root of the sample size= 34.4/SQRT(60)= 4.441021. We then substute the SEM into the formula we get (300-276.1)/4.441021 =5.38165.  This z-score has a 1-sided above probability of less than .00001. In other words, it would be extremely rare given a sample of 60 students for the mean to be higher than 300. The reason it is so rare is that 60 is a large enough sample that it should be pretty close to the population mean of 276.1.

h)      Are you more likely to randomly select one student with a test score greater than 300, or are you more likely to select a sample of students with a mean test score greater than 300? Explain?

a.       The way to find out is compare the z-score we got in the last portion, which is from a mean, with a z-score from just 1 score. That z-score is (300-276.1)/34.4 = .69476. This z-score has a 1-sided probability of .2436 or 24.36%. So comparing that to the probability of getting a MEAN score above 300 of less than 1%, it's clear that it is more common to see an individual score higher. The reason is that as your sample size increases the mean of that sample will converge onto the population mean, which is 276.1. With just one student, its like having a sample size of 1, and it is more likely for a single point to vary further from the population mean.