MATH 1530

Unit 3 Normal Distributions and Confidence Intervals

5.2 Normal Distributions: Finding Probabilities

Writing your probability statement:

Example \(P(x>10)=P(z>.4136)=\)

Use the following formula to calculate the z scores for your x values: \(z=\frac{x-\mu}{\sigma}\)

Labeling your bell curve:

You will be required to demonstrate your understanding that the probability calculated on the Normal Curve is equal to the corresponding probability calculated on the Standard Normal Curve.

• Label your bell curve with two rows of information on the horizontal axis.

  a.  z score values   Use μ=0 and σ=1.

  b.  x values   Use given μ and σ.

• Shade the probability area. Verify the shaded region matches your probability statement for x and z.

Calculating your probability:

Writing your final statement:

If one __________ is randomly selected, the probability the/their __________ is lessthan/greaterthan/between____________is approximately equal to ______.

1. The distribution of IQ scores is normally distributed with a mean of 100 and a standard deviation of 15. An Oregon couple claimed their two children were taken away from them because they scored too low on an IQ test. She scored 72 and he scored 66. Find the probability that a randomly selected IQ score will be below 73

   \(\mu = \) 100

   \(\sigma = \) 15

   \(z_{73}=\frac{73-100}{15}=-1.8\)

   \(P(x \lt 73)=P(z \lt -1.8)=0.0359\)

   If one person’s IQ score is randomly selected, the probability their score is less than 73 is approximately equal to 0.0359 or 3.59%.

2. Are the IQ scores of the couple unusual? (Hint: Calculate z-score for each.)

\(z_{72}=\frac{72-100}{15}=-1.87\)

\(z_{66}=\frac{66-100}{15}=-2.27\)

Her IQ is not unusual, but his is unusual.

3. To determine the average wait time at Taco Bell in the Miami area, 16 graduate students randomly selected 16 taco bell stores. Each student visited a Taco Bell store at 12:15 pm and ordered a crunchy beef taco, a burrito supreme, and a soda. The wait time to receive the meal was recorded for each restaurant. The distribution is normal with a mean of 22.4 minutes and a standard deviation of 6.2 minutes.
   1. If one Taco Bell is randomly selected, what is the probability the wait time will be more than 20 minutes?

      \(z=\frac{20-22.4}{6.2}=-0.3871\)

      \(P(x>20)=P(z>-0.3871)=.6506\)

      If one Taco Bell is randomly selected, the probability the student's wait time was more than 20 minutes is approximately equal to 0.6506 or 65.06%.

   2. If one Taco Bell is randomly selected, what is the probability the wait time will be between 25 minutes and 35 minutes?

      \(z_{25}=\frac{25-22.4}{6.2} \approx 0.4194\)

      \(z_{35}=\frac{35-22.4}{6.2} \approx 2.0322\)

      \(P(25 \lt x \lt 35)=P(.419 \lt z \lt 2.0322)=.3164\)

      If one Taco Bell is randomly selected, the probability the student's wait time was between 25 and 35 minutes is approximately equal to 0.3164 or 31.64%.