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Finding Z – scores & Normal Distribution

Finding Z – scores & Normal Distribution. Using the Standard Normal Distribution Week 9 Chapter’s 5.1, 5.2, 5.3. Normal Distribution.

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Finding Z – scores & Normal Distribution

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  1. Finding Z – scores& Normal Distribution Using the Standard Normal Distribution Week 9 Chapter’s 5.1, 5.2, 5.3

  2. Normal Distribution Normal Distribution - is a very important statistical data distribution pattern occurring in many natural phenomena, such as height, blood pressure, grades, IQ, baby birth weights, etc. Normal Curve - when graphing the normal distribution as a histogram, it will create a bell-shaped curve known as a normal curve. It is based on Probability! You’ll see!

  3. Normal Distribution Curve:

  4. What is this curve all about? The shape of the curve is bell-shaped The graph falls off evenly on either side of the mean.  (symmetrical) 50% of the distribution lies on the left of the mean 50% lies to the right of the mean. (above) The spread of the normal distribution is controlled by the standard deviation. The mean and the median are the same in a normal distribution. (and even the mode)

  5. Features of Standard Normal Curve • Mean is the center • 68% of the area is within one S.D. • 95% of area is within two S.D.’s • 99% of area is within 3 S.D.’s • As each tail increases/decreases, the graph approaches zero (y axis), but never equals zero on each end. • For each of these problems we will need pull-out table IV in the back of text

  6. What is a Z – Score? • Z-score’s allow us a method of converting, proportionally, a study sample to the whole population. • Z-Score’s are the exact number of standard deviations that the given value is away from the mean of a NORMAL CURVE. • Table IV always solves for the area to the left of the Z-Score!

  7. Finding the area to the left of a Z (Ex. 1) – Find the area under the standard normal curve that lies to the left of Z=1.34.

  8. Finding the area to the right of a Z (Ex. 2) - Find the area under the standard normal curve that lies to the right of Z = -1.07.

  9. Finding the area in-between two Z’s (Ex. 3) - Find the area under the standard normal curve that lies between Z=-2.04 and Z=1.25.

  10. Formula: x = data value u = population mean

  11. Practice examples: For each of the following examples, Look for the words "normally distributed" in a question before using Table IV to solve them. Don’t forget - Table IV always solves for the area to the left of the Z-Score!

  12. Finding Probabilities The shaded area under the curve is equal to the probability of the specific event occurring. • Ex (4) - A shoe manufacturer collected data regarding men's shoe sizes and found that the distribution of sizes exactly fits a normal curve.  If the mean shoe size is 11 and the standard deviation is 1.5. (a)What is the probability of randomly selecting a man with a shoe size smaller than 9.5? (b)If I surveyed 40 men, how many would be expected to wear smaller than 9.5?

  13. How did we get that answer: This is how many SD’s from the mean • -1.00 is a Z-score (# of S.D.’s from the mean) that refers to the area to the left of that position. Find it in Table IV. • -1.00 = .1587 • We want the area to the left of that curve, so, this is the answer. Table IV gives us the answer for area to the left of the curve. • (b) .1587(40) = 6.3 = 6

  14. Ex (5) – Gas mileage of vehicles follows a normal curve. A Ford Escape claims to get 25 mpg highway, with a standard deviation of 1.6 mpg. A Ford Escape is selected at random. (a) What is the probability that it will get more than 28 mpg? (b) If I sampled 250 Ford Escapes, how many would I expect to get more than 28 mpg?

  15. How did we get that answer: • 1.875 is a Z-score (# of S.D.’s from the mean) that refers to the area to the left of that position. • 1.875 = .9696 • We want the area to the right of that curve, thus • 1- .9696 = .0304

  16. Ex (6) – This past week gas prices followed a normal distribution curve and averaged $3.73 per gallon, with a standard deviation of 3 cents. What percentage of gas stations charge between $ 3.68 and $ 3.77?

  17. Ex (7) – This week gas prices followed a normal distribution curve and averaged $3.71 per gallon, with a standard deviation of 3 cents. • What percent of stations charge at least $3.77? • What percent will charge less than $3.71? • What percent will charge less than $3.69? • What percent will charge in-between $3.67and $3.75 per gallon? • If I sampled 30 gas stations, how many would charge between $3.67and $3.75 per gallon?

  18. NOW – Going back from probabilities to Z-Scores: Chapter 5.3 – Finding Z-scores from probabilities – Transforming a Z-score to an X-value • Look up .9406 on Table 4 • What Z-score corresponds to this area?

  19. Finding Z-score’s of area to the left (Ex. 8) • Find the Z-score so that the area to the left is 10.75% (b) Find the Z-score that represents the 75th percentile? (c) Find the Z-score so that the area to the left is .88 (d) Find the Z-score so that the area to the left is .9880

  20. Transforming a Z-score to an x-value Look for the three ingredients to solve for x: Population mean, standard deviation, and you will need the Z-score that corresponds to the given percent (or probability) Try: 90th percentile

  21. Finding Z-score’s of area to the left (Ex. 9) – The national average on the math portion of the SAT is a 510 with a standard deviation of 130. SAT scores follow a normal curve. (a) What score represents the 90th percentile? (b) What score will place you at the 35th percentile?

  22. Finding Z-score’s of area to the Right (Ex. 10) – Find the Z score so that the area under the standard normal curve to the right is .7881

  23. Find the Z score of area to the Right (Ex. 11) – A batch of Northern Pike at a local fish hatchery has a mean length of a 8 inches just as they are released to the wild. Their lengths are normally distributed with a standard deviation of 1.25 inches. What is the shortest length that could still be considered part of the top 15% of lengths?

  24. Find the Z score of area in-between two Z’s (Ex. 12) – Find the Z score that divides the middle 90% of the area under the standard normal curve.

  25. Critical Two-tailed Z value: - Used to find the remaining percent on the outside of the area under the curve. • 90% is equal to 1-.90=.10 • .10/2 = .05 = 5%

  26. M&M Packaging Ex (13) – A bag of M&M’s contains 40 candies with a Standard deviation of 3 candies. The packaging machine is considered un-calibrated if it packages bags outside of 80%, centered about the mean. What interval must the candies be between for sale?

  27. The Central Limit Theorem: • As the sample sizes increases, the sampling distribution becomes more accurate in representation of the entire population. • Thus, As additional observations are added to the sample, the difference of the Sample mean and the population mean approaches ZERO. ( No difference)

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