Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.

a.The current speed limit is 65 mph. What is the proportion of vehicles less than or equal

to the speed limit?

b.What proportion of the vehicles would be going less than 50 mph?

- A new speed limit will be initiated such that approximately 10% of vehicles will be over

the speed limit. What is the new speed limit based on this criterion?

- In what way do you think the actual distribution of speeds differs from a normal distribution?

Ans.

- m= 71 mph; s = 8 mph

The proportion of vehicles less than or equal to the speed limit 65 mph = 0.2266

- The proportion of the vehicles would be going less than 50 mph = 0.0043

- The new speed limit would be 81.25 mph (approximately 81 mph).

- The actual distribution of speeds on a highway differs from the normal distribution in that

- It is not symmetric. The speeds are not symmetrically distributed about the central vaue. Two speeds that are an equal amount greater or lower than the mean speed, may not have the same frequency as is required for a normal distribution.

- Secondly, the speeds further away from the average speed do not drop in frequency as rapidly as in normal distribution. The frequency of the extreme values may not be as low as in normal distribution, resulting in a fatter tail. The distribution may also be skewed in one direction. This is especially true in highways where the distribution will be expected to be positively skewed- greater frequency in the higher speed range than lower.

- A group of students at a school takes a history test. The distribution is normal with a mean of 25, and a standard deviation of 4. (a) Everyone who scores in the top 30% of the distribution gets a certificate. What is the lowest score someone can get and still earn a certificate? (b) The top 5% of the scores get to compete in a statewide history contest. What is the lowest score someone can get and still go onto compete with the rest of the state?

Ans. For the test, the mean score m= 25; s = 4

Using the normal calculator (Ch-7, Lane),

- the lowest score someone can get and still earn a certificate = 27.096 (or approx.. 27.1)

- the lowest score someone can get and still go onto compete with the rest of the state = 31.58 (or approx. 31.6)

- Use the normal distribution to approximate the binomial distribution and find the probability of getting 15 to 18 heads out of 25 flips. Compare this to what you get when you calculate the probability using the binomial distribution. Write your answers out to four decimal places.

**Ans ****For the binomial distribution** of coin flipping, probability of getting a head (success) in one flip = p = 0.5

N = no. of trials = 25

Mean is m = N*p = 25 * 0.5 = 12.5

Variance σ^{2} = Np(1-p) = (25)(0.5)(0.5) = 6.25

The standard deviation is therefore σ = √6.25 = 2.5.

Using the binomial calculator (Lane CH-5),

the **probability of getting 15 to 18 heads out of 25 flips = 0.2049**

**Using the normal distribution to approximate the binomial distribution**: The area between 14.5 and 18.5 in the normal distribution curve is an approximation of the probability of obtaining 15 to 18 heads.

Area below 14.5 = 0.7881

Area below 18.5 = 0.9918

Therefore,

Area between 14.5 and 18.5 = 0.9918 – 0.7881 = 0.2037

So in this case,

the **probability of getting 15 to 18 heads out of 25 flips = 0.2037**

Illowsky CH-6

- The patient recovery time from a particular surgical procedure is normally distributed with a mean of 5.3 days and a standard deviation of 2.1 days.

What is the median recovery time?

- 2.7

- 5.3

- 7.4

- 2.1

Ans. b. 5.3

In the normal distribution, mean and median are the same.

- Height and weight are two measurements used to track a child’s development. The World Health Organization measures child development by comparing the weights of children who are the same height and the same gender. In 2009, weights for all 80 cm girls in the reference population had a mean μ = 10.2 kg and standard deviation σ = 0.8 kg. Weights are normally distributed. X ~ N(10.2, 0.8). Calculate the z-scores that correspond to the following weights and interpret them.

- 11 kg

b.7.9 kg

c.12.2 kg

Ans. a. z-score for 11 kg = (11- μ)/ σ = (11- 10.2)/ 0.8 = 0.8/0.8 = 1

Interpretation: The weight 11 kg is 1 standard deviation above the average weight value of 10.2 kg.

- z-score for 7.9 kg = (7.9 – μ)/ σ = (7.9 – 10.2)/ 0.8 = -2.3/0.8 = -2.875

Interpretation: The weight 7.9 kg is 2.875 standard deviations below the average weight value of 10.2 kg.

- z-score for 12.2 kg = (12.2- μ)/ σ = (12.2 – 10.2)/ 0.8 = 2.0/0.8 = 2.5

Interpretation: The weight 12.2 kg is 2.5 standard deviations above the average weight value of 10.2 kg. As the z-score for this weight is greater than 2, it can be concluded that 12.2 kg weight for a girl 8 cm tall is an “unusual” value.

- Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet.

a.If X = distance in feet for a fly ball, then X ~ _____(_____,_____)

b.If one fly ball is randomly chosen from this distribution, what is the probability that this ball traveled fewer than 220 feet? Sketch the graph. Scale the horizontal axis X. Shade the region corresponding to the probability. Find the probability.

c.Find the 80th percentile of the distribution of fly balls. Sketch the graph, and write the probability statement.

Ans.a. If X = distance in feet for a fly ball, then X ~ _N____(__250___,__50___)

- z-score for 220 ft = (220- μ)/ σ = (220 – 250)/ 50 = -30/50 = -0.6

Area to the left of z= -0.6 will represent the probability that this ball traveled fewer than 220 feet.

Area to the left of -0.6 = 0.5 – 0.22575 = 0.27425

So the probability that this ball traveled fewer than 220 feet is **0.27425**.

- The 80th percentile of the distribution of fly balls = 292.073 feet

80% of the fly balls travelled fewer than 292.073 feet. 20% of the fly balls travelled greater than 292.073 feet.

- Facebook provides a variety of statistics on its Web site that detail the growth and popularity of the site.

On average, 28 percent of 18 to 34 year olds check their Facebook profiles before getting out of bed in the morning. Suppose this percentage follows a normal distribution with a standard deviation of five percent.

a.Find the probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30.

b.Find the 95th percentile, and express it in a sentence.

Ans. If X= percentage of 18 to 34 year old who check their Facebook profiles before getting out of bed in the morning, then X~ N(28, 5).

m= 28; s = 5

- z-score for 30% = (30- m)/s = (30-28)/5 = 2/5 = 0.4

The area to the right of z-score 0.4 will represent probability that the percent of 18 to 34-year-olds who check Facebook before getting out of bed in the morning is at least 30.

From the normal table (Illowsky pg 849),

Area to the right of 0.4 = 1- (0.5 + 0.15542) = 0.3446

The required probability is 0.3446.

- 95% of the area will be below that percentage.

Area to the left = 0.95

From the normal table we need to find the area value closest to 0.95 – 0.5 = 0.45 and get the corresponding z-score. The closest match lower than 0.45 that we can get on the table is 0.44950 and the corresponding z-score value is 1.64.

Now we convert the z-score 1.64 to the x value using the formula

z * s = x – m

x = z * s + m = 1.64 *5 + 28 = 36.2

So the 95^{th} percentile is 36.2%.

It means the probability that 36.2 percent of the 18 to 34-year-olds who check Facebook before getting out of bed in the morning, is 0.95 or less.

CH-7 Illowsky

- Suppose that the distance of fly balls hit to the outfield (in baseball) is normally distributed with a mean of 250 feet and a standard deviation of 50 feet. We randomly sample 49 fly balls.

- If X ̄= average distance in feet for 49 fly balls, then X ̄ ~ __()

- What is the probability that the 49 balls traveled an average of less than 240 feet? Sketch the graph. Scale the horizontal axis for X ̄ . Shade the region corresponding to the probability. Find the probability.

- Find the 80th percentile of the distribution of the average of 49 fly balls.

Ans. a. X ̄ ~ _N_(250, 50/√49)

Or **X ̄~ N(250, 7.1429)**

- z-score for 240 = (240-250)/7.1429 = -10/7.1429 =-1.4

Probability that the 49 balls traveled an average of less than 240 feet = area left of z score -1.4 = 0.5 – 0.41924 = **0.0808**

- 80th percentile of the distribution of the average of 49 fly balls =
**256.01 (using normal calculator, Lane)**

Calculations using the z score table, Illowsky:

Area 0.80 (= 0.5 + 0.3) corresponds to z-score 0.84

X ̄ = z* s_{x} + m_{x }= 0.84 * 7.1429 + 250 = **256.00**

- Which of the following is NOT TRUE about the distribution for averages?

- The mean, median, and mode are equal.

- The area under the curve is one.

- The curve never touches the x-axis.

- The curve is skewed to the right.

Ans. d. The curve is skewed to the right.

- A typical adult has an average IQ score of 105 with a standard deviation of 20. If 20 randomly selected adults are given an IQ test, what is the probability that the sample mean scores will be between 85 and 125 points?

Ans.

m_{x }= m = 105

s_{x }= s/√n = 20/ √20 = 4.4721

If X ̄ = average IQ score for 20 adults,

X ̄ **~ N(105, 4.4721)**

** **

z-score for 85 = (85-105)/4.4721 =-20/4.4721 = -4.4721

Area o the left of z score -4.4721 = 0

z-score for 125 = (125-105)/4.4721 = 20/4.4721 = 4.4721

Area to the left of z-score 4.4721 = 1

**Probability that the sample mean scores will be between 85 and 125 points **

= Area between z scores -4.4721 (mean score 85) and +4.4721 (mean score 125)

= Area to the left of z-score 4.4721 – Area to the left of z-score -4.4721

= 1- 0

= **1**

^{ }