Finding Z

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Finding Z scores Normal Distribution

• Using the Standard Normal Distribution
• Week 9
• Chapters 5.1, 5.2, 5.3

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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! Youll see!

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Normal Distribution Curve
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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)

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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

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What is a Z Score?

• Z-scores allow us a method of converting,
  proportionally, a study sample to the whole
  population.
• Z-Scores 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!

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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 Z1.34.

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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.

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Finding the area in-between two Zs

• (Ex. 3) - Find the area under the standard normal
  curve that lies between Z-2.04 and Z1.25.

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Formula

• x data value
• u population mean

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Practice examples

• For each of the following examples, Look for the
  words "normally distributed" in a question before
  using Table IV to solve them.
• Dont forget - Table IV always solves for the
  area to the left of the Z-Score!

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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?

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How did we get that answer
This is how many SDs 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

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• 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?

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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

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• 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?

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• 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?

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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?

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Finding Z-scores 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

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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
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Finding Z-scores 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?

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Finding Z-scores 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

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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?

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Find the Z score of area in-between two Zs

• (Ex. 12) Find the Z score that divides the
  middle 90 of the area under the standard normal
  curve.

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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

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MM Packaging

• Ex (13) A bag of MMs 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?

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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)