A company services home air conditioners. It is known that...

A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes. a. What is the probability that a single service call takes more than 65 minutes? b. What is the probability that a single service call takes between 50 and 70 minutes? c. The probability is 0.025 that a single service call takes more than how many minutes? d. Find the shortest range of times that includes $50 \%$ of all service calls. e. A random sample of four service calls is taken. What is the probability that exactly two of them take more than 65 minutes?

A company services home air conditioners. It is known that times for service calls follow a normal distribution with a mean of 60 minutes and a standard deviation of 10 minutes.
a. What is the probability that a single service call takes more than 65 minutes?
b. What is the probability that a single service call takes between 50 and 70 minutes?
c. The probability is 0.025 that a single service call takes more than how many minutes?
d. Find the shortest range of times that includes $50 \%$ of all service calls.
e. A random sample of four service calls is taken. What is the probability that exactly two of them take more than 65 minutes?

Key Concepts

Normal Distribution

This is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and is defined by two parameters: the mean (which indicates the center of the distribution) and the standard deviation (which measures the spread or dispersion of the data).

Standardization

Standardization is the process of converting a normal random variable with any mean and standard deviation into a standard normal variable (Z) with a mean of 0 and a standard deviation of 1. This technique facilitates the calculation of probabilities using standard normal distribution tables or functions, as it allows any normal distribution problem to be tackled using a universal framework.

Cumulative Distribution Function (CDF)

The cumulative distribution function is used to determine the probability that a random variable takes on a value less than or equal to a specific value. In the context of the normal distribution, the CDF enables the computation of the probability for a range of values by referencing standard normal tables or using statistical software.

Percentiles

Percentiles are values below which a given percentage of observations in a data set fall. Finding a percentile in a normal distribution involves locating the corresponding Z-score that satisfies the desired cumulative probability, which can then be transformed back to the original scale using the mean and standard deviation.

Central Interval

A central interval in a probability distribution is one that is symmetric about the mean and contains a specified proportion of the distribution's probability, such as 50%. Determining this interval involves finding two cutoff points that equally balance the tail probabilities outside the interval.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes (success or failure) with constant probabilities. In the context of service times, it can be used to calculate the probability of a specific number of events occurring (such as service calls taking more than a certain time) in a given sample size.

Transcript

0:00 The question is worded.

00:01 The probability is 0 .025 that a single call will take more than how many minutes.

00:06 So we want to find this value.

00:08 And we have to find out how many standard deviations above what that z value is.

00:13 And so i'm going to actually look up, this is either 0 .975 this way, or you could look up 0 .025 and then just have to take the opposite of that value.

00:23 But 0 .975 corresponds with 1 .96.

00:28 And so we need to take roughly two standard deviations.

00:33 And so we'll take that 60 and add on 1 .96, again, roughly 2 times 10.

00:39 And so 60 plus 1 .96 times 10 is going to end up giving us 79 .6 minutes.

00:49 So approximately 80 minutes.

00:53 Now part d, we want to find part d what the shortest interest.

00:58 That includes 50%, and that shortest interval that includes 50%, it's going to be the one that is symmetrically located around the mean.

01:08 And so we need to find if this z value here is 0 .25 below, 0 .25 above, and we need to find what these endpoints are.

01:19 So if i look up 0 .25, 0 .25 in the body of this, and i'm at 0 .2 .2.

01:29 It is right in between 0 .67, negative .6 .7 and negative 0 .68.

01:38 It's a tiny bit closer to this one...