Question

An elevator has a placard stating that the maximum capacity is 1610 lb---10 passengers.​ So, 10 adult male passengers can have a mean weight of up to

1610 divided by 10 = 161 pounds. If the elevator is loaded with 10 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than

161lb.​ (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 35 lb.)

Does this elevator appear to be​ safe?

The probability the elevator is overloaded is?

​(Round to four decimal places as​ needed.)

Answer 1

Concepts and reason

Normal distribution: Normal distribution is a continuous distribution of data that has the bell-shaped curve. The normally distributed random variable x has mean and standard deviation.

Also, the standard normal distribution represents a normal curve with mean 0 and standard deviation 1. Thus, the parameters involved in a normal distribution are mean and standard deviation.

Standardized z-score: The standardized z-score represents the number of standard deviations the data point is away from the mean.

• If the z-score takes positive value when it is above the mean (0).

• If the z-score takes negative value when it is below the mean (0).

Sampling distribution of sample mean:

The sampling distribution of the sample mean for the given sample size n consists of the collection of the means of all possible samples of size n from the population.

Fundamentals

Let , then the standard z-score is found using the formula given below:

The sample distribution of the sample mean follows normal distribution with mean, and standard deviation, . That is, .

The z score for the sampling distribution is,

Where X denotes the individual raw score, denotes the population mean, and denotes the population standard deviation.

Procedure for finding the z-value is listed below:

1.From the table of standard normal distribution, locate the probability value.

2.Move left until the first column is reached.

3.Move upward until the top row is reached.

4.Locate the probability value, by the intersection of the row and column values gives the area to the left of z.

Conditions of unusualness for probability value:

• If , then it is unusual.

• If , then it is usual.

The probability the elevator is overloaded is obtained as shown below:

From the information given, the weights of males are normally distributed with a mean of 165 lb and a standard deviation of 35 lb, and the sample size is 10 adult male passengers can have a mean weight. That is, .

Let X denotes the weight of elevator.

The required probability is,

From the “standard normal table”, the area to the left of $z=-0.36 $ is 0.3594.

The elevator appears to be safe or not is obtained as shown below:

The probability the elevator is overloaded is 0.6406.

It is clear that the probability value is greater than 0.05, this indicates that the elevator is not appear to be safe.

Ans:

The probability the elevator is overloaded is 0.6406.

The elevator is not appearing to be safe.