1. “Find the probability that a randomly-selected healthy adult has a body temperature…”
less than 96.9oF
greater than 97.6oF
between 98.0oF and 99.0oF
less than 99.6oF or greater than 100.6oF
differs from population mean by more than 1.0oF
Method #1 – by changing the x problem into a z problem and finding the area left/right of z value
Method #2 – by using TI-84 with x values directly and let the calculator handle details of the z
You still need to draw pictures – you just have to.

2. Let’s do this by CONVERTING the x-PROBLEM
Example 6.12: Finding the Probability that a Normally Distributed Random Variable Will Be Less Than a Given Value
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature less than 96.9 °F?
Let’s do this by CONVERTING the x-PROBLEM
into a z-PROBLEM AND SEEING ALL THE DETAILS.
Note that it’s a _________ distribution.
Mean μ = ________
Standard deviation σ = ___________
Convert x = ________ to z.
Formula: 𝒛= 𝒙−𝝁 𝝈
z =

3. Draw a picture! Two axes. Draw a picture – label axes. .
Convert x = 96.90
into a z value – use formula
𝑧= (𝑥−𝜇) 𝜎
Draw a picture – label axes. .
Convert mean μ = 98.60
into a z value – easy!
Mean of Standard Normal Distribution
is always exactly what?
TWO
AXES
x-axis
and
z-axis z

4. The picture leads us to the solution.
The area to the left of z = _______. (Use either the printed table or normalcdf(low z, high z) to find it.) So the area to the left of x = ______ is also ________. Since Area Is Probability, the probability that a healthy adult has a body temperature less than ______o is ________, or about ______ %

5. TI-84 normalcdf talks in both z and x language
This is what we learned last time.
In z-language: normalcdf(low z, high z) gives you the area, that is, the probability
In x-language, you can give it the x values, but tell it the mean and standard deviation, too: normalcdf(low x, high x, μ, σ) This gives you the area, that is, the probability, without having to change the x into z yourself. The calculator does x-to-z conversions transparently.
NEW!

6. Example 6.12: Finding the Probability that a Normally Distributed Random Variable Will Be Less Than a Given Value (cont.)
Previous example REPEATED but using the x-language version of the TI-84’s normalcdf : We want the area between negative infinity and We use −1× for −∞. (-) 1 2ND COMMA 9 9
SUPERB SHORTCUT !!!
The details of changing the x problem to a z-problem are handled inside the TI-84, as long as we tell it the mean & std. deviation for this x.

7. Example 6.12 again, using Excel
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature less than 96.9 °F? =NORM.DIST(x value, mean, standard dev., TRUE)

8. Example 6.13: Finding the Probability that a Normally Distributed Random Variable Will Be Greater Than a Given Value
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature greater than 97.6 °F?
If we do the conversion to z ourselves:
x = 97.6 converts to z = _______
Make the sketch and find area (it’s on the next slide)
𝒛= 𝒙−𝝁 𝝈
Conclusion (in a complete sentence in plain English):

9. mean of 98. 60 °F and a standard deviation of 0. 73 °F
mean of °F and a standard deviation of 0.73 °F. …probability … greater than 97.6 °F?
On x-axis, label mean & std dev & also label 97.6.
On z-axis, label 0 at mean and label the z-value.
Shade under curve (which side?)
Use z table to find area. . x z

10. Conclusion (in a complete sentence):
Example 6.13: Finding the Probability that a Normally Distributed Random Variable Will Be Greater Than a Given Value
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature greater than 97.6 °F? Same problem but this time with TI-84 shortcut:
Conclusion (in a complete sentence):

11. Example with Excel
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature greater than 97.6 °F? Use 1-NORM.DIST(x, mean, stdev, TRUE))
Why is this 1– here in the front?

12. Example 6.14: Finding the Probability that a Normally Distributed Random Variable Will Be between Two Given Values
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature between 98 °F and 99 °F?
Doing it with explicit conversion to z: (PICTURE IN NEXT SLIDE – WRITE ON IT)
Convert x = 98 to z = ________ Convert x = 99 to z = ________.
The area __________ z = _____ and z = ______ is _________________,
and that’s the probability of temp between ______ and ______.

13. Between 98 and 99oF . x z

14. Same problem but with TI-84 normalcdf(low,high,mean,stdev)
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature between 98 °F and 99 °F? Do directly with TI-84 and state plain conclusion:

15. Example 6.14 with Excel
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature between 98 °F and 99 °F? NEED TO SUBTRACT TWO NORM.DIST() VALUES

16. Example 6.15: Finding the Probability that a Normally Distributed Random Variable Will Be in the Tails Defined by Two Given Values
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature less than 99.6 °F or greater than °F? (Sketch is needed, see at right →!) Recall method: 1 – area between. State conclusion plainly:

17. less than 99.6 °F or greater than 100.6 °F?x z

18. Label mean and the low and high endpoints.
Example 6.16: Finding the Probability that a Normally Distributed Random Variable Will Differ from the Mean by More Than a Given Value
Body temperatures of adults are normally distributed with a mean of °F and a standard deviation of 0.73 °F. What is the probability of a healthy adult having a body temperature that differs from the population mean by more than 1 °F? SKETCH FIRST – on next slide →
Label mean and the low and high endpoints.
Shade darkly the segments that correspond to the area we want to find.
TI-84 command (using the complement trick): normalcdf( ______, ______, ______, ______)
Conclusion:

19. Differs from population mean by more than 1 °F?. x z