1.On a test the mean is 75 and the standard deviation is 5.

A. What is the probability that a randomly selected score is higher than 80?
B. What percentage of the students scored higher than 80?
C. What percentage of the scores were less than 68?
D. What score separates the top 20% from the rest?
E. What score separates the bottom 15% from the rest?
F. If a set of 32 students are selected, what is the probability that the sample mean is less than 77?

A, B, C. Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores you've found.

D, E. Reverse process from the same table. Use the % to find the Z score, then insert into the formula to find the score.

F. For Distribution of means, Z = (score-mean)/SEm (Standard Error of the mean)

SEm = SD/√(n-1)

Use same table.

Here is a great normal distribution calculator

http://davidmlane.com/hyperstat/z_table.html

You can enter the data either directly or as z-scores.

Thanks

population scores with a mean (u) of 200 and a standard deviation (Q) of 10 wiht a normal distribution what score would cut off 5 percent os scores

To solve these questions, we will be using the Z-score formula, which calculates the number of standard deviations a particular value is away from the mean. The Z-score formula is:

Z = (X - μ) / σ

Where:
Z = z-score
X = the value we are interested in
μ = mean of the distribution
σ = standard deviation of the distribution

We will also be using a standard normal distribution table (also known as a Z-table) to find the probabilities associated with different Z-scores.

Now, let's solve each question step by step:

A. What is the probability that a randomly selected score is higher than 80?

To find the probability of a score being higher than 80, we need to calculate the Z-score for 80 using the given mean and standard deviation. Then we can use the Z-table to find the corresponding probability.

Z = (80 - 75) / 5 = 1

Looking up the Z-score of 1 in the Z-table, we find that the probability is 0.8413.

Therefore, the probability that a randomly selected score is higher than 80 is 0.8413 (or 84.13%).

B. What percentage of the students scored higher than 80?

To find the percentage of students who scored higher than 80, we can use the same Z-score of 1 we calculated in the previous question.

Using the Z-table, we can find that the probability of a score being less than 80 (since we want the percentage above 80) is 0.8413.

Since we want the percentage above 80, we subtract this probability from 1:
1 - 0.8413 = 0.1587

Therefore, 15.87% of the students scored higher than 80.

C. What percentage of the scores were less than 68?

To find the percentage of scores less than 68, we need to calculate the Z-score for 68 using the given mean and standard deviation and then use the Z-table to find the corresponding probability.

Z = (68 - 75) / 5 = -1.4

Looking up the Z-score of -1.4 in the Z-table, we find that the probability is 0.0808.

Therefore, the percentage of scores that were less than 68 is 0.0808 (or 8.08%).

D. What score separates the top 20% from the rest?

To find the score that separates the top 20% from the rest, we need to find the Z-score corresponding to this percentile and then convert it back to a raw score using the Z-score formula.

Looking up the Z-score that corresponds to the top 20% in the Z-table, we find it to be approximately 0.8416.

Setting up the Z-score formula, we have:

0.8416 = (X - 75) / 5

Solving for X, we get:

(X - 75) = 0.8416 * 5
X - 75 = 4.208
X = 79.208

Therefore, the score that separates the top 20% from the rest is approximately 79.21.

E. What score separates the bottom 15% from the rest?

To find the score that separates the bottom 15% from the rest, we need to find the Z-score corresponding to this percentile and then convert it back to a raw score using the Z-score formula.

Looking up the Z-score that corresponds to the bottom 15% in the Z-table, we find it to be approximately -1.036.

Setting up the Z-score formula, we have:

-1.036 = (X - 75) / 5

Solving for X, we get:

(X - 75) = -1.036 * 5
X - 75 = -5.18
X = 69.82

Therefore, the score that separates the bottom 15% from the rest is approximately 69.82.

F. If a set of 32 students are selected, what is the probability that the sample mean is less than 77?

The sample mean follows a normal distribution, and in this case, we can use the Central Limit Theorem to assume that the sample mean will also have mean 75 and a standard deviation of (original standard deviation) / sqrt(n), where n is the sample size. Therefore, the new standard deviation is 5 / sqrt(32) approximately 0.882.

To calculate the probability that the sample mean is less than 77, we need to calculate the Z-score for this value using the new mean and standard deviation and then use the Z-table to find the corresponding probability.

Z = (77 - 75) / 0.882 = 2.271

Looking up the Z-score of 2.271 in the Z-table, we find that the probability is approximately 0.9885.

Therefore, the probability that the sample mean is less than 77 is approximately 0.9885 (or 98.85%).

To recap:
A. The probability that a randomly selected score is higher than 80 is 0.8413 (84.13%).
B. The percentage of students who scored higher than 80 is 0.1587 (15.87%).
C. The percentage of scores that were less than 68 is 0.0808 (8.08%).
D. The score that separates the top 20% from the rest is approximately 79.21.
E. The score that separates the bottom 15% from the rest is approximately 69.82.
F. The probability that the sample mean is less than 77 is approximately 0.9885 (98.85%).