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submit in groups of either 1 or 2 or 3. _____________________________ Turn in 1 copy per group. _____________________________ 1) The area under a probability density curve must sum to 1 (see p. 85). True or False ß (.5 pt) circle the correct answer (.5 pt) Explain your answer: The Context: The ACT exam is scored using only whole numbers. In 2010, the mean ACT score of all those who took the test was 21 and the standard deviation was 4.7. 2) a. (.5 pt) Compute the z-score for an ACT score of 23. b. (.5 pt) In Ch 2 of our text, we computed z-scores and interpreted then. Interpret the z-score you found in part a. (Hint: it refers the position of the ACT score relative to the mean score.) Interpretation: 3) (1 pt) ACT scores are not a continuous variable. Why? Even though ACT scores are not continuous, we can use the normal distribution to approximate the distribution of ACT scores by assuming that the distribution is normal and that the number of scores is large enough so our approximation makes sense. To compute the probability that a randomly selected student gets at or below certain ACT score, we can use the normal distribution. To start, we have to assume that the distribution of ACT scores is approximately normal. Then we can determine the z-score for the certain ACT score. In class, we then relied on the table of values in the z-table (pp. 288–298 in your text and also posted in Moodle) to determine the probability we were asked for. In this project you will learn how to use your calculator instead of the z-tables.

4) (1 pt) Use your z-tables and your answer to #2 to help you determine the probability that a randomly selected student gets at or below an ACT score of 23. (Note that on the next problem you will find this answer with your calculator and compare your answers.)

• Include an appropriate sketch. • Refer to your previous answers (do not repeat work) • Express your answer using appropriate probability notation.

Instead of using a z-table to help you determine probabilities, you can use functions in your Ti– 83/84 calculator. Remember that a standard normal distribution (i.e., a distribution of z-scores) always has a mean of 0 (µ = 0) and a standard deviation of 1 (σ = 1). Ti-84 instructions for finding P(z ≤ z1) [Consider “z1” as a specific z-score]

1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd Understand that the lower bound is – ∞ and the upper bound is “z1”.

[Note: use very large values, like –999999 and +999999, for bounds that are infinite.] 3rd enter in: lower bound,upper bound, µ,σà it should look something like

normalcdf(–999999,z1,0,1)

5) (1 pt) Use the normcdf function on your calculator and your z-score answer to #2 to determine the probability that a randomly selected student gets at or below an ACT score of 23.

• Include what you entered in your calc. and its result. • Refer to your previous answers (do not repeat work) • Express your answer using appropriate probability notation.

Your calculators are so powerful that you can use them to find the probability from #4 (or #5) without first converting to z-scores! To do this you will again use the normcdf function on your calculator. Ti-84 instructions for finding P(x ≤ x1) [Consider “x1” as a specific x value, like an ACT score]

1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd enter in: lower bound,upper bound, µ,σà it should look something like normalcdf(–999999,x1, µ,σ), where µ is the mean of the population and σ is the st dev of the population

Recall: The mean ACT score of all those who took the test in 2010 was 21 and the standard deviation was 4.7

6) (1 pt) Without converting to z-scores, use the normcdf function on your calculator to determine the probability that a randomly selected student gets at or below an ACT score of 23.

• Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation. • Write your final answer using words, in the context of the problem.

7) (1 pt) Compare your answers to #4, #5, and #6. They are all computations of the same probability. a) Are they all the same result? ______ b) How close are they to each other? c) Which result do you think is the most accurate? In the previous problems (4 – 6), you investigated the probability of randomly selecting a value (from a normal distribution) that is less than or equal to a given value (i.e., the probability of choosing an ACT score of 23 or less). Now let’s investigate the probability of randomly selecting a value (from a normal distribution) that is greater than a given value (i.e., the probability of choosing an ACT score higher than 23). Let’s work with z-scores first: Recall you previously found P(z ≤ z1) [Consider “a” as a specific z-score] Now to find P(z > z1) use the fact that P(z ≤ z1) + P(z > z1) = 1 because z ≤ z1 and z > z1 together make up all possibilities for z, i.e., they are compliments. So, if you first find P(z ≤ z1) , you can then compute P(z > z1) = 1 – P(z ≤ z1). 8) (1 pt) Use your z-tables to help you determine the probability of randomly selecting a z-score that is greater than z = 1.05

• Include an appropriate sketch. • Show steps using appropriate probability notation. • Express your answer using appropriate probability notation.

Next, instead of using a z-table to help you determine probabilities, you can use functions in your Ti–83/84 calculator. Using your calculator you can directly find P(z > z1).

Ti-84 instructions for finding P(z > z1) [Consider “z1” as a specific z-score] 1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd Understand that the lower bound is “z1” and the upper bound is +∞.

[Note: use very large values, like –999999 and +999999, for bounds that are infinite.] 3rd enter in: lower bound,upper bound, µ,σà it should look something like

normalcdf(z1,999999,0,1)

9) (1 pt) Re-compute the probability you found in #8 by using your calculator to directly find it, i.e., do not use the compliment. Find the probability of randomly selecting a z-score that is greater than z = 1.05

• Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation.

Again, your calculators are so powerful that you can use them to find the probabilities without first converting to z-scores! To do this you will again use the normcdf function on your calculator.

Ti-84 instructions for finding P(x > x1) [Consider “x1” as a specific x value, like an ACT score] 1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd enter in: lower bound,upper bound, µ,σà it should look something like normalcdf(x1,999999,µ,σ), where µ is the mean of the population and σ is the st dev of the population

Recall: The mean ACT score of all those who took the test in 2010 was 21 and the standard deviation was 4.7

10) (1 pt) Without converting to z-scores, use the normcdf function on your calculator to determine the probability that a randomly selected person gets above an ACT score of 23.

• Include an appropriate sketch. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation. • Write your final answer using words, in the context of the problem.

Now let’s investigate the probability of randomly selecting a value (from a normal distribution) that is between two given values (i.e., the probability of choosing an ACT score greater than or equal to 23 but less than or equal to 25).

Let’s work with z-scores first. Recall you previously found P(z ≤ z1) [Consider “a” as a specific z-score] Now suppose that z2 is greater than z1 (z1 < z2). To find P(z1 ≤ z ≤ z2) you will need to subtract: P(z ≤ z2) – P(z ≤ z1) = P(z1 ≤ z ≤ z2). 11) (1 pt) Use your z-tables to help you determine the probability of randomly selecting a z- score that is greater than or equal to z = – 2.98 but less than or equal to z = 1.05. (Note that you can use your results from #9 to reduce your work.)

• Include an appropriate sketch. • Show steps using appropriate probability notation. • When appropriate, refer to your previous answers (do not repeat work). • Express your answer using appropriate probability notation.

Next, instead of using a z-table to help you determine probabilities, you can use functions in your Ti–83/84 calculator. By following the steps you used in #10, you could solve #11 by converting to z-scores and then finding probabilities. However, you can also use your calculator to skip some steps, i.e., your calculator can directly find P(z1 ≤ z ≤ z2).

Ti-84 instructions for finding P(z1 ≤ z ≤ z2)

1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd Understand that the lower bound is “z1” and the upper bound is “z2”. 3rd enter in: lower bound,upper bound, µ,σà it should look something like

normalcdf(z1,z2,0,1)

12) (1 pt) Re-compute the probability you found in #11. Use your calculator to directly find the probability of randomly selecting a z-score that is greater than or equal to z = – 2.98 but less than or equal to z = 1.05.

• Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation.

Again, your calculators are so powerful that you can use them to find the probabilities without first converting to z-scores! To do this you will again use the normcdf function on your calculator.

Ti-84 instructions for finding P(x1 ≤ x ≤ x2) 1st Select the normcdf function: 2nd à VARS(DIST) à 2:normalcdf( 2nd enter in: lower bound,upper bound, µ,σà it should look something like normalcdf(x1,x2,µ,σ), where µ is the mean of the population and σ is the st dev of the population

Recall: The mean ACT score of all those who took the test in 2010 was 21 and the standard deviation was 4.7 13) (1 pt) Without converting to z-scores, use the normcdf function on your calculator to determine the probability of randomly choosing a person with an ACT score greater than or equal to 23 but less than or equal to 25.

• Include an appropriate sketch. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation. • Write your final answer using words, in the context of the problem.

14) (extra credit up to +1) Optional Challenge Problem: Use your Ti calculator to compute the following probability (don’t forget the rules of probability from Ch 4).

P(z < –2.56 or z > 1.39) = ? • Include an appropriate sketch. • Show steps. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation.

Finally, we can also reverse this process you have been using to solve #’s 2 – 13. That is, given a probability (area under the normal curve) you can determine the corresponding z value or corresponding x value. 15) (1 pt) Use the z-table to find the z-score such that randomly choosing that z-score or less has a probability of 0.0017.

• Include an appropriate sketch. • Use the z-table to find the score that corresponds to the given probability. • Express the information you found in the z-table using appropriate probability notation. • Give your final answer.

As you now may already suspect, you can use your calculator can solve #15 without using the z- table. You will use the invNorm function on your Ti-83/84. Ti-84 instructions for finding z1, such that P(z ≤ z1) = A [Consider “A” as a specific value for a probability]:

1st Select the invNorm function: 2nd à VARS(DIST) à 2: invNorm( 2nd Understand that the value of A is the probability (or area) that corresponds to the z- score you are finding. 3rd enter in: probability (or area),µ,σà it should look something like invNorm(A,0,1)

16) For each problem below, find the corresponding z-score(s). Remember that the invNorm function only works with areas (probabilities) at or below a specific z-score.

• Include an appropriate sketch. • Show steps. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation.

a.) (.5 pt) Find z1 if P(z ≤ z1) = 0.0017

b.) (.5 pt) Find z1 if P(z > z1) = 0.7337

c.) (.5 pt) Find –z1 and z1 if P(–z1 < z < z1) = 0.1976

d.) (.5 pt) Find z1 if P(0 < z < z1) = 0.3595

So now, if you are given a problem with context (like ACT scores) and if you are given a probability then you should be able to find corresponding x-values. As before we can do much of this on your calculator.

Recall: The mean ACT score of all those who took the test in 2010 was 21 and the standard deviation was 4.7.

17) (1 pt) Using z-scores and the invNorm function on your calculator (but not the z-table) find the lowest 20% of ACT scores. (Remember that in these types of contexts, probability, percentage and proportion are all equivalent.)

• Include an appropriate sketch. • Show steps. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation. • Write your final answer using words, in the context of the problem.

OK, now let’s use your calculator to skip the z-score step.

Ti-84 instructions for finding x1, such that P(x ≤ x1) = A [Consider “A” as a specific value for a probability]

1st Select the invNorm function: 2nd à VARS(DIST) à 2: invNorm( 2nd Understand that the value of A is the probability (or area) that corresponds to the x value you are finding. 3rd enter in: probability (or area),µ,σà it should look something like invNorm(A,µ,σ)

Recall: The mean ACT score of all those who took the test in 2010 was 21 and the standard deviation was 4.7. Remember that the invNorm function only works with areas (probabilities) at or below a specific x-value.

18) Using the invNorm function on your calculator (without using z-scores) answer the following questions. (Remember that in these types of contexts, probability, percentage and proportion are all equivalent.)

• Include an appropriate sketch. • Show steps. • Include what you entered in your calc. and its result. • Express your answer using appropriate probability notation. • Write your final answer using words, in the context of the problem. a) (1 pt) Find the lowest 20% of ACT scores.

b) (1 pt) Find the highest 15% percent of ACT scores.

c) (1 pt) Find the middle 40% of ACT scores.

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