Question

Find the area of the shaded region. The graph to the right depicts IQ scores of? adults, and those scores are normally distributed with a mean of 100 and a standard deviation of 15.

the shaded area reads 90

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 (0).

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

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.

Instructions to build the shaded area and p-value is given below:

1.In EXCEL, Select Add-Ins > MegaStat > Probability.

2.Choose Continuous probability distributions.

3.Select Normal distribution and select calculate P given z and enter z as ____.

4.Enter mean as ____ and standard deviation as ___.

5.Click Ok.

The z-score for the shaded area X equals 90 is calculated as follows:

Let X denotes the IQ scores of adults which follows normal distribution with mean 100 and the standard deviation 15. That is .

The required z-score is,

The probability area for the shaded region is obtained as follows:

Instructions to build the shaded area and p-value is given below:

1.In EXCEL, Select Add-Ins > MegaStat > Probability.

2.Choose Continuous probability distributions.

3.Select Normal distribution and select calculate P given z and enter z as -0.67.

4.Enter mean as 0 and standard deviation as 1.

5.Click Ok.

Follow the above instructions to get the shaded are and p-value in the output as shown below:

From the EXCEL output the p-value for the dark grey region is shaded area until 90 is 0.2514.

Ans:

Thus, the probability of shaded area reads until 90 is 0.2514