Unit 2.7: Corequisite

The speeds of vehicles driven between 6 pm and 8 pm on part of a highway follow a distribution that is approximately normal. The mean vehicle speed is 68 miles per hour. The standard deviation is 6.5 miles per hour. What proportion of vehicle speeds are less than 60 miles per hour? Round to 2 decimal places.

0.11. P(x < 60) = P(Z < -1.23) ≈ 0.1093

(1) What is the probability that a randomly selected U.S. adult spends less than 2 hours watching television each day? Round to 2 decimal places. This problem can be written as P(x < 120) = P(Z < -1.11) = ?

0.13

The speed limit on this part of the highway is 75 miles per hour. What is the probability that a randomly selected vehicle is traveling faster than the speed limit? Enter as a decimal and round to 2 decimal places.

0.14. P(x > 75) = P(Z > 1.08) ≈ 0.1401.

What is the Z-score that has an area of 0.10 to its right? Round the Z-score to 2 decimal places.

1.28.

June temperatures in the Southwestern U.S. are approximately normal with mean 68.0 °F and standard deviation 2.3 °F. 6) Suppose we want to know the June temperature in the Southwestern U.S. that separates the bottom 30%. Use technology to find the corresponding Z-score. Round to 2 decimal places. It may help to sketch the distribution and think about where the separating score would be. Then, use technology to find the Z-score.

-0.52 The location of this Z-score is shown below. The area to the left is 0.30. The area to the right is 0.70. The image below displays the command in Desmos that finds a Z-score with an area of 0.30 to its left.

What is the Z-score that has an area of 0.20 to its left? Round the Z-score to 2 decimal places.

-0.84.

(2) What is the probability that a randomly selected U.S. adult spends more than 3 hours watching television each day? Round to 2 decimal places. This problem can be written as P(x > 180) = P(Z > 0.47) = ?

0.32

June temperatures in the Northeastern U.S. are approximately normal with mean 64 °F and standard deviation 2.0 °F. Suppose we want to know the June temperature in the Northeastern U.S. that separates the top 15% from the lower 85%. Use technology to find the corresponding Z-score. Round to 2 decimal places.

1.04 The image below displays the command in Desmos that finds a Z-score with an area of 0.85 to its left.

A vehicle speed has a Z-score of -1.5 in this distribution. What is the vehicle speed? Round to the nearest mile per hour.

58 miles per hour. x = (68) + (-1.5)(6.5) = 58.25, which rounds to 58 miles per hour.

How fast is a vehicle traveling if its speed is in the slowest 25% of vehicle speeds? Round to one decimal place.

63.6. Z = -0.67; The corresponding speed is 63.6 miles per hour. A speed of 63.6 miles per hour or slower is in the bottom 25% of the speeds.

(5) Convert the Z-score to a June temperature in the Northeastern U.S. Round to the nearest degree.

66°F x = (64) + (1.04)(2.0) = 66.08°F, so 66°F rounded to the nearest degree.

(7) Convert the Z-score to a June temperature in the Southwestern U.S. Round to the nearest degree.

67°F x = (68) + (-0.52)(2.3) = 66.8°F, so 67°F rounded to the nearest degree.

Normal distributions model continuous random variables that have bell-shaped and symmetric distributions. The standard normal distribution can be used to find probabilities associated with a range of values in any normal distribution. Technology or tables can help find probabilities related to ranges of Z-scores.

Calculate probabilities using normal distributions. Find critical values in the standard normal distribution. Determine the cutoff values in a normal distribution.

For the remainder of this lesson, we will explore monthly temperature distributions for two regions of the United States based on temperature data from 1895 - 2017. A monthly temperature is the average temperature of an entire region over a 1-month period.

June temperatures in the Northeastern U.S. are approximately normal with mean 64.0 °F and standard deviation 2.0 °F. June temperatures in the Southwestern U.S. are approximately normal with mean 68.0 °F and standard deviation 2.3 °F. The normal curves below model these temperature distributions.

How fast is a vehicle traveling if its speed is in the fastest 5% of vehicle speeds? Round to one decimal place.

Solution 78.7. Z = 1.64; 78.7 miles per hour or more.

3) What amounts of daily television watching should be considered unusual based on this distribution? Explain. Recall that unusual values in normal distributions are values that are two standard deviations or more away from the mean.

Unusual values are values that are less than or equal to 86 minutes, or values that are greater than or equal to 238 minutes.

(8) Which is more unlikely: randomly observing a June temperature in the Northeastern U.S. that is greater than 68 °F, or randomly observing a June temperature in the Southwestern U.S. that is less than 64 °F? Justify your answer.

We can compare Z-scores or probabilities. The Z-score of 68 °F in the Northeastern U.S. distribution is 2.0. The Z-score of 64 °F in the Southwestern U.S. distribution is -1.73. The Z-score of 68 °F in the Northeastern U.S. distribution is farther from the mean, so randomly observing a June temperature as extreme as this would be more unlikely.

The American Time Use Survey tracks the amount of time U.S. adults spend sleeping, doing leisure activities, socializing and communicating, performing household activities, and doing other activities. In 2016, the survey found that U.S. adults spend, on average, 2.7 hours (162 minutes) each day watching television. Americans spend more time watching television than any other leisure activity.

We will assume that the amount of television U.S. adults watch each day is approximately normal with a mean of 162 minutes and a standard deviation of 38 minutes. In each problem below, x represents the amount of minutes a random U.S. adult spends watching television each day. Compute the corresponding probability.

normal distributions model

random variables whose probability distributions are symmetric and bell shaped.

We can determine a cutoff value (a value in a distribution that separates a certain percentage of values from the remainder of the distribution) by first determining the Z-score that separates the same percentage. Once the Z-score is determined, we use the formula: x=μ+Zσ

to convert Z to a value, x, from a normal population that has a mean μ and standard deviation σ.