Let’s recap what we know how to do: We can use normalcdf to find the area under the normal curve between two z-scores. We proved the empirical rule this way. We can use invnorm to find the z-score for a given area to the left. We can use normalcdf and invnorm with raw scores instead of z-scores if we also provide a mean and standard deviation. We used the concept of a percentile for applications with the normal distribution.

The words used in the problem will guide you as to which calculator function (normalcdf or invnorm) to use. And the problem provides raw scores or z- scores, If the problem asks for area, proportion, # of data values, percentile, or probability normalcdf Some number between 0 and 1 that represents area, a proportion, or a percentile. And the problem provides area, a percentile, proportion, or probability If the problem asks for a score(s) (either z-score or raw scores) invnorm Some score, either a raw score or a z-score. input output

And now, we’ll observe yet another application of the normal curve : probability. PROBABILITY = PROPORTION under the normal curve Recall the probability distribution for the number of minutes it takes a student to get to class.

Remember, to find the probability a student will require from 5 to 7 minutes to get to the next class we added the areas corresponding to the shaded rectangles associated with X = 5 minutes, X = 6 minutes, and X = 7 minutes.

But for continuous distributions that are modeled by the normal curve, we no longer have rectangles, but a smooth bell curve. To find that probability that a student takes between 5 and 7 minutes to get to class, we calculate the area under the normal curve between 5 and 7.

In general, the probability of an Event E in a normal distribution is defined to be: P(Event E) = Proportion of area under the normal curve satisfying Event E.

Assume the distribution of the playing careers of major league baseball players can be approximated by a normal distribution with a mean of 8 years and a standard deviation of 4 years. Find the probability that a player selected at random will have a career that will last: (a) Less than 6 years (b) Longer than 14 years (c) Between 4 and 10 years

Ask yourself, does this problem give us area and ask for scores, (we’ll use invnorm) or does this problem give scores and asks for area (we’ll use normalcdf)? Since the problem is asking for probability (which is an application of area under the normal curve) and provides scores (“less than 6, longer than 14,” etc.) we will be using normalcdf to answer the probability question.

(a) Find the probability that a player selected at random will have a career that will last less than 6 years Start by labeling a normal curve with the mean and standard deviation. (mean = 8, standard deviation = 4) Locate the raw score in question, and shade the appropriate portion of the curve.

(a) Find the probability that a player selected at random will have a career that will last less than 6 years The proportion of area less than 6 is the area to the left of 6. normalcdf(-E99,6,8,4) = The probability that a player’s career will last less than 6 years is

(b) Find the probability that a player selected at random will have a career that will last more than 14 years Start by labeling a normal curve with the mean and standard deviation. (mean = 8, standard deviation = 4) Locate the raw score in question, and shade the appropriate portion of the curve.

(b) Find the probability that a player selected at random will have a career that will last more than 14 years The proportion of area more than 14 is the area to the right of 14. normalcdf(14, E99,8,4) = The probability that a player’s career will last more than 14 years is

(c) Find the probability that a player selected at random will have a career that will last between 4 and 10 years Start by labeling a normal curve with the mean and standard deviation. (mean = 8, standard deviation = 4) Locate the raw scores in question, and shade the appropriate portion of the curve.

(c) Find the probability that a player selected at random will have a career that will last between 4 and 10 years The proportion of area between than 4 and 10 is found by: normalcdf(4,10,8,4) = The probability that a player’s career will last between 4 and 10 years is

Suppose the systolic blood pressure for an adult male above age 55 is normally distributed with a mean of 145 and a standard deviation of 20. If a sixty year old man is selected at random, find the probability that his blood pressure is: (a) Between 140 and 160? (b) Greater than 170? (c) If 4000 adults males above age 55 have their blood pressure checked, how many would you expect to have blood pressures less than 130?

(d) Volunteers from a Red Cross mobile unit that is checking blood pressures decides to recommend for additional medical diagnosis only those men about age 55 whose blood pressures are at the third quartile. What would be the minimum systolic blood pressure required to be recommended for additional medical diagnosis?

Pg:415