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Applications of the Normal distribution

Applications of the Normal distribution. Remember……. The z-score is found by using the following formula: Z = value – mean standard deviation. To solve these problems we will have to:. Transform the values of the variable to z values and then find the areas under the curve

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Applications of the Normal distribution

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  1. Applications of the Normal distribution

  2. Remember…… • The z-score is found by using the following formula: • Z = value – mean • standard deviation

  3. To solve these problems we will have to: • Transform the values of the variable to z values and then find the areas under the curve • We will do this by finding the z-score

  4. example • A survey found that women spend on average $146.21 on beauty products during the summer. Assume the standard deviation is $29.44. Find the percentage of women who spend less than $160.00. Assume the variable is normally distributed. • Find the z-score. • Value= mean= SD= • 2. Find the area using Table E.

  5. Sometimes drawing the graph will help us find our area.

  6. An American household generates an average of 28 pounds of paper garbage each month. Assume the standard deviation is 2 pounds. • a) Find the probability of generating between 27 and 31 pounds per month.

  7. An American household generates an average of 28 pounds of paper garbage each month. Assume the standard deviation is 2 pounds. • b) Find the probability of generating more than 30.2 pounds per month.

  8. We can also use a normal distribution to answer the question “how many?” • Americans consume an average of 1.64 cups of coffee per day. Assume the variable is approximately normal with a standard deviation of 0.24 cup. If 500 people are selected, about how many will drink less than 1 cup per day? • Step 1: Draw your figure • Step 2: Find the z-value for 1 • Step 3: Find the area to the left of your z-value. • Step 4: Multiply the sample size by your area.

  9. Americans consume an average of 29 pounds of French fries in a year. Assume that the variable is normal with a standard deviation of 2. If 200 people are selected, approximately how many will eat less than 24.5 pounds of fries per year?

  10. Finding data values given specific probabilities • We will have to “work backwards.” • Step 1: Find the area under the curve using your percentage. (you may need to draw a graph) • Step 2: Find the z-value that corresponds to the area (if the exact area isn’t there, find the closest z-value.) • Step 3: Substitute into the formula and solve for X: • Z = X – mean • standard deviation • Your answer is the value that X must have.

  11. Example • To qualify for the police academy, candidates must score in the top 10% on a general abilities test. The test has a mean of 200 and a standard deviation of 20. Find the lowest possible score a person must score to qualify.

  12. Example 2 • A researcher wishes to select people in the middle 60% of the population based on blood pressure. If the mean systolic blood pressure is 120 and the standard deviation is 8, find the upper and lower reading that would qualify people to participate in the study.

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