In a study of the length of time it takes to play Major...

In a study of the length of time it takes to play Major League Baseball games during the early 2008 season, the variable "time of game" appeared to be normally distributed, with a mean of 2 hours 49 minutes and a standard deviation of 21 minutes. a. Some fans describe a game as "unmanageably long" if it takes more than 3 hours. What is the probability that a randomly identified game was unmanageably long? b. Many fans describe a game lasting less than 2 hours, 30 minutes as "quick." What is the probability that a randomly selected game was quick? c. What are the bounds of the interquartile range for the variable time of game? d. What are the bounds for the middle $90 \%$ of the variable time of game?

In a study of the length of time it takes to play Major League Baseball games during the early 2008 season, the variable "time of game" appeared to be normally distributed, with a mean of 2 hours 49 minutes and a standard deviation of 21 minutes.
a. Some fans describe a game as "unmanageably long" if it takes more than 3 hours. What is the probability that a randomly identified game was unmanageably long?
b. Many fans describe a game lasting less than 2 hours, 30 minutes as "quick." What is the probability that a randomly selected game was quick?
c. What are the bounds of the interquartile range for the variable time of game?
d. What are the bounds for the middle $90 \%$ of the variable time of game?

Key Concepts

Central Confidence Intervals

Central confidence intervals, such as the middle 90% interval, are used to indicate the range of values within which a specified percentage of the data falls. In the context of a normal distribution, these intervals are determined by identifying the corresponding quantiles that leave equal tails outside the interval, providing a measure of the variability and reliability of the estimates.

Interquartile Range (IQR)

The interquartile range is a measure of statistical dispersion, which shows the range within which the middle 50% of the data lie. It is determined by calculating the difference between the third quartile (75th percentile) and the first quartile (25th percentile), and is useful for understanding the spread of the central portion of the data distribution.

Probability Calculations Using the Normal Distribution

Probability calculations using the normal distribution involve determining the area under the normal curve for a given interval. By converting the values of interest to Z-scores, one can compute the probability that a normally distributed variable falls within a particular range or exceeds a certain threshold.

Standardization and Z-scores

Standardization is a process that involves converting a normal random variable to a standard normal variable (Z-score), which has a mean of 0 and a standard deviation of 1. This transformation simplifies probability calculations by allowing the use of standard normal tables or cumulative distribution functions to determine the probabilities of events occurring within specified ranges.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, with data near the mean more frequent in occurrence than data far from the mean. It is characterized by its mean and standard deviation, and many real-world phenomena are modeled using this distribution due to the central limit theorem.

Transcript

00:01 In this question, we're presented with a study that finds that the length of major league baseball games during a season was normally distributed with a mean of two hours in 49 minutes and a standard deviation of 21 minutes.

00:14 So we have a normal distribution with the mean.

00:18 So two hours is 120 minutes plus 49 minutes.

00:23 That's two hours and 49 minutes.

00:25 That is 169 minutes.

00:29 And a standard deviation of 21 minutes.

00:35 And in part a, we're asked, what is the probability that a game exceeds three hours? so that is the probability that x is greater than 180 minutes.

00:47 And we can convert to z scores using this conversion.

00:50 Z is equal to x minus mu over sigma.

00:56 So this is equal to the probability that z is greater than 180 minus 169 divided by 21, which is equal to the probability that z is greater than 0 .524.

01:18 And that comes out to a probability of 0 .30.

01:23 So the probability that a game exceeds three hours is 0 .3 .00.

01:30 For part b, we're asked, what is the probability that a game is less than 2 and a half hours? so that's the probability that x is less than 150 minutes.

01:41 And that is the probability that z is less than 0 .905.

01:51 And that comes out to 0 .183.

01:54 So we can say there is a probability of 0 .183 that a game lasts less than 2 .5 hours.

02:04 For part c, we're asked to find the interquartile range.

02:07 So the interquartile range is defined by these boundaries, q1 and q3.

02:14 Q1 is the first quartile.

02:16 Q3 is the third quartile.

02:19 So just to remind you, so if we have our distribution, for time of game for major league baseball games.

02:27 The mean is 169 minutes, and the standard deviation is 21 minutes.

02:33 So the first quartile is the point or the time for which 25 % of distribution is to the left.

02:45 And conversely, the third quartile is the point at which 25 % is to the right, or 75 % is to the left of it.

02:57 So we can see that the probability that x is less than q1 is equal to 0 .25.

03:11 So that's how we define the boundary of the first quartile.

03:19 So let's solve for q1.

03:21 So that's the probability that x is less than q1 is equal to 0 .25.

03:33 And we can also say that this is the probability that z is less than q1 minus 100.

03:43 169 divided by 21.

03:51 And the probability that z is less than a value equals 0 .25.

03:59 So we can go to our tables or a calculator or software to find that this corresponds with z is equal to minus 0 .675...

Please give Ace some feedback