Week 7 lecture_math_221_dec_2012
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This document discusses normal distributions and how to calculate probabilities and confidence intervals related to normal distributions using Minitab software. Key topics covered include the standard normal distribution, using Minitab to calculate normal distribution probabilities, examples of finding z-scores and areas under the normal curve, confidence intervals for means and proportions, and interpreting confidence intervals.
Week 7 lecture_math_221_dec_2012
- 1. B. Heard (This material can not be copied or posted without the author’s consent. Students may download one copy for personal use.)
- 2. Standard Normal Distribution The “standard” normal distribution is a normal distribution with mean zero and where the standard deviation (and variance) equals one. The Total Area under the curve is one (1) or 100% (This is true for all normal distributions regardless of the mean and standard deviation).
- 3. Using Minitab for Normal Distribution calculations. Use Calc >> Probability Distributions >> Normal Examples Follow
- 4. Example Theaverage fish in Happy Lake weighs 2 pounds with a standard deviation of 0.5 pounds. If Bob catches a fish that weighs 3.2 pounds. What could you say about the catch?
- 5. Since this is the Cumulative Distribution Function, it “fills” from left to right. Therefore, you could say his catch was in the “Top 1 %”
- 6. Example Theaverage fish in Happy Lake weighs 2 pounds with a standard deviation of 0.5 pounds. If Bob catches a fish that weighs 1.35 pounds. What could you say about the catch?
- 7. Since this is the Cumulative Distribution Function, it “fills” from left to right. Therefore, you could say his catch was in the “Bottom 10 %”
- 8. Other types of questions If you have a normal distribution with a mu = 100 and sigma = 15, what number corresponds to a z = -2 -2 = (x – 100)/15 Multiply both sides by 15 to get -30 = x – 100 Add 100 to each side to get 70 = x So “70” is my answer, I just did a little Algebra.
- 9. Another type of question Say we take 120 samples of size 81 each from a distribution we know is normal. Calculate the standard deviation of the sample means if we know the population variance is 25. (Answer next chart)
- 10. Answer The Central Limit Theorem tells us the variance is the Population variance divided by the Sample Size. We can just take the square root to get the standard deviation. Variance = 25/81 or 0.309 Standard Deviation = Square Root(25/81) = 5/9 = 0.556
- 11. Findingz scores Example The area to the left of the “z” is 0.6262. What z score corresponds to this area. Use Calc >> Probability Distributions >> Normal (Set Mean = 0 and Standard Deviation to 1 and use “INVERSE Cumulative Probability”
- 12. Answer is 0.322 rounded to three decimals. Remember the distribution fills from left to right.
- 13. Another type of question In a normal distribution with mu = 40 and sigma = 10 find P(32 < x < 44) Easy, but this takes a couple of steps. Using Calc >> Probability Distributions >> Normal find the probabililties that x < 32 and x < 44 using the Cumulative Probability option.
- 14. Continued Get results for Both 32 and then 44.
- 15. Answer Subtract 0.655422 – 0.211855 To get 0.443567 Or 0.444 rounded to three decimals P(32 < x < 44) = 0.444 based on the given mean and std deviation.
- 16. Confidence Intervals and Examples Charts follow
- 17. Interpreting Confidence Intervals If you have a 90% confidence interval of (15.5, 23.7) for a population mean, it simply means “There is a 90% chance that the population mean is contained in the interval (15.5, 23.7) It’s really that simple.
- 18. Finding Confidence Intervals A luxury car company wants to estimate the true mean cost of its competitor’s automobiles. It randomly samples 180 of its competitors sticker prices. The mean cost is $65,000 with a standard deviation of $3200. Find a 95% confidence interval for the true mean cost of the competitor’s automobiles. Write a statement about the interval.
- 19. It randomly samples 180 of its competitors sticker prices. The mean cost is $65,000 with a standard deviation of $3200. Find a 95% confidence interval… Use Stat >> Basic Statistics >> 1 sample Z Make sure to click Options and set to 95%
- 21. Click your OK buttons… Confidence Interval is (64533, 65467), which means we can be 95% confident the true mean cost of the competitor’s vehicles are between those two values.
- 22. FindConfidence Intervals of Proportions Example A student wants to estimate what proportion of the student body eats on campus. The student randomly samples 200 students and finds 120 eat on campus. Using a 95% confidence interval, estimate the true proportion of students who eat on campus. Write a statement about the confidence level and interval.
- 23. Example Solution p hat = 120/200 = 0.60 q hat = 1- 0.60 = 0.40 n p hat = 200 * 0.60 = 120 n q hat = 200 * 0.40 = 80 Using E = Zc* Square Root ((p hat * q hat)/n) = 1.96 * Square Root ((0.60*0.40)/200) =0.0679 Now we subtract this from the mean for the left side of the interval and add it to the mean for the right side. (0.60 – 0.0679, 0.60 + 0.0679) = (0.5321, 0.6679) So with 95% confidence, we can say the population proportion of students who eat lunch on campus is (0.5321, 0.6679) or between 53.21% and 66.79%.