Question

A normal distribution has a mean of μ = 30 and a standard deviation of = 12. For the score, X = 18, indicate whether the body is to the right or left of the score and find the proportion of the distribution located in the body.

Answer 1

In a normal distribution with a mean of 30 and a standard deviation of 12, a score of X = 18 is to the left of the mean. The proportion of the distribution located in the body to the left of X = 18 is approximately 15.87%.

Step-by-step explanation:

In a normal distribution, if a score is to the left of the mean, it has a negative z-score. If a score is to the right of the mean, it has a positive z-score. In this case, we are given a score of X = 18 in a normal distribution with a mean of μ = 30 and a standard deviation of σ = 12.

To find the proportion of the distribution located in the body to the left of X = 18, we need to calculate the z-score for this score.

The z-score formula is: z = (X - μ) / σ.

Plugging in the values, we get: z = (18 - 30) / 12 = -1.

Since the z-score is negative (-1), it means that the score of X = 18 is to the left of the mean.

To find the proportion of the distribution located in the body to the left of X = 18, we can use a standard normal distribution table or a calculator to find the corresponding area under the curve. The area to the left of a z-score of -1 is approximately 0.1587, which means about 15.87% of the distribution is located in the body to the left of X = 18.

Answer 2

For the score X = 18 in a normal distribution with a mean
`\(\mu = 30\)` and a standard deviation
`\(\sigma = 12\)`, the body is to the left of the score. The proportion of the distribution located to the left of X = 18 is approximately 0.3446.

Step-by-step explanation:

In a normal distribution, the mean
`\(\mu\)` represents the central tendency, while the standard deviation
`\(\sigma\)` measures the dispersion or spread of the data points. To determine whether the body of the distribution is to the left or right of the score X = 18, we'll use z-scores. The z-score formula is
`\(z = (X - \mu)/(\sigma)\)`, where X is the score,
`\(\mu\)` is the mean, and
`\(\sigma\)` is the standard deviation.

Substituting the given values into the formula, the z-score for X = 18 in this scenario is
`\(z = (18 - 30)/(12) = -1\)`. A negative z-score indicates that the score X = 18 is to the left of the mean in a normal distribution.

To find the proportion of the distribution to the left of X = 18, we refer to the standard normal distribution table or use statistical software. The table provides the cumulative probability for the corresponding z-score, which is approximately 0.3446 for a z-score of -1. This value represents the proportion of the distribution located to the left of the score X = 18 in the given normal distribution with a mean of 30 and a standard deviation of 12.