Question

Assume a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable x is normally distributed.

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 meanand 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

• If the z-score takes negative value when it is below the mean

Fundamentals

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

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

Formula to find the probability of X is greater than a and less than b is,

Let X is a random variable that denotes the braking distance (in feet) follows normal distribution with parametersand.

From the give information, a bell shaped curve is given. The mean of the barking distance is 168 and standard deviation is 5.54. That is, $u=168 $ and$o=5.54 $. The lower value of x is 160 and upper value of x is 168.

The z-score when x is 160:

The z-score when x is 168:

The probability that the member selected at random is from the shaded area of the graph is obtained below:

From the information, for$x=168 $the z-score is 0 and $x=160 $the z-score is –1.44.

The required probability is,

From the “standard normal table”, the area to the left of is 0.5 and the area to the left of$z=-1.44 $ is 0.0749.

Ans:

Thus, the probability that the member selected at random is from the shaded area of the graph is 0.4251.