Example 1. Find the x value that has a left tail area of.1131 P ( x <??? ) =. 1131

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1 Section 6 4D: Finding a Value of x with a Given tail arae Label the shaded area for both graphs. Find the value for z and label the z axis. Find the value for x for the given area under the normal curve with the specified mean and standard deviation. Round the value for x to 2 decimal places. Given: µ x = 3.2 and Example Find the x value that has a of.3 P ( x <??? ) =. 3 If the for the x value is.3 then the for the z curve is also.3 =.3 =.3 x <??? µ x = 3.2 z <??? Find a score that has a as close to.3 as possible. Negative Scores Standard Normal () Distribution: Cumulative Area to the LEFT of If =.2 then convert z to x by the following formula x = µ + z(σ ) setup for finding x: x = µ + z(σ ) x = 3.2.2(.45) = 2.66 x = 2.66 P ( x < 2.66 ) =.3 About % of the x values are less than 2.66 Stat 3 6 4D Lecture Page of 5 22 Eitel

2 Example 2 Given: µ x = 2.9 and σ x = 3.4 Find the x value that has a right tail area of.34 P ( x >??? ) =. 34 If the right tail area for the If the for the x distribution is.34 x distribution is.8686 then the then the is.8686 for the z curve is also.8686 =.8686 right tail area =.34 =.8686 µ x = 2.9 σ x = 3.4 x =??? Find a score that has a as close to.8686 as possible. Positive Scores Standard Normal () Distribution: Cumulative Area to the LEFT of If =.2 then convert z to x by the following formula x = µ + z(σ ) setup for finding x: x = µ + z(σ ) x = (3.4) = 25.7 x = 25.7 P ( x > 25.7 ) =. 34 About 3% of the x values are more than than 25.7 Stat 3 6 4D Lecture Page 2 of 5 22 Eitel

3 Example 3 Given: µ x = 92.6 and σ x = 7. Find the x value that has a right tail area of.9382 P ( x >??? ) = If the right tail area for the If the for the x distribution is.9382 x distribution is.68 then the then the is.68 for the z curve is also.68 =.68 right tail area =.9382 =.68 x =??? µ x = 92.6 σ x = 7. x Find a score that has a as close to.8686 as possible. Negative Scores Standard Normal () Distribution: Cumulative Area to the LEFT of If =.54 then convert z to x by the following formula x = µ + z(σ ) setup for finding x: x = µ + z(σ ) x = (7.) = 8.67 x = 8.67 P ( x > 8.67 ) = About 94% of the x values are more than than 8.67 Stat 3 6 4D Lecture Page 3 of 5 22 Eitel

4 Example 4 Kradle to Kindergarten Daycare Center knows that the ages of the children it cares for are normally distributed with a mean age of 3.2 years and a standard deviation of.45 years. The daycare center wants to mail out an advertisement to the parents of the youngest 35% of its members. What is the age that separates the youngest 35% of the children from the older children? Given: µ x = 3.2 and Find the x value that has a of.35 P ( x <.??? ) =. 35 If the for the x value is.35 then the for the z curve is also.35 =.35 =.35 x =??? µ x = 3.2 Find a score that has a as close to.35 as possible. Negative Scores Standard Normal () Distribution: Cumulative Area to the LEFT of If =.39 then convert z to x by the following formula x = µ + z(σ ) setup for finding x: x = µ + z(σ ) x = (.45) = 3.24 x = 3.2 If I send an advertisement to the parents of children less than 3.2 years old, the advertisement will be sent to the youngest 35% of the daycare children. Note: The age that separates the youngest 35% of the children from the older children is called P 35. We use a Normal Distribution and the Standard Normal Distribution to find P 35 instead of the sorted list and percentile formula technique used in the past. We do not have the values listed for the distribution of x values so we cannot sort the list and find the location of P 35. Stat 3 6 4D Lecture Page 4 of 5 22 Eitel

5 Example 5 Kradle to Kindergarten Daycare Center knows that the ages of the children it cares for are normally distributed with a mean age of 3.2 years and a standard deviation of.45 years. The daycare center wants to send out an add to the parents of the oldest 5% of its members. What is the age that separates the oldest 5% of the children from the younger children? Given: µ x = 3.2 and Find the x value that has a right tail area of.5 P ( x <.??? ) =.5 If the right tail area for the x values is.5 then the is.5 =.85 If the for the x values is.85 then the or the z values is also.85 left tail area =.85 right tail area =.5 =.85 µ x = 3.2 x =??? If the for the x values is.85 then the or the z values is also.85 Find a score that has a as close to.85 as possible. Positive Scores Standard Normal () Distribution: Cumulative Area to the LEFT of If =.4 then convert z to x by the following formula x = µ + z(σ ) setup for finding x: x = µ + z(σ ) x = (.45) = x = 3.67 If I send an add to the parents of children more than 3.67 years old, the add will be sent to the oldest 5% of the daycare children. Note: The age that separates the oldest 5% from the youngest 5% of the children from the older children is called P 5. Stat 3 6 4D Lecture Page 5 of 5 22 Eitel