Question

The following data is from a simple random sample: 15, 23, x, 37, 19, 32. If the point estimate of the population mean is 23, then the value of x is:

• A. 12
• B. 30
• C. 21
• D. 24

Hint:

When calculating the sample mean, always ensure that you correctly sum all the data points and divide by the number of data points. If a value is missing (like in this case \( x \)), use the given mean to set up an equation and solve for the unknown. Multiplying both sides by the number of data points can help eliminate the denominator and simplify the equation for easier solving.

Answer 1

The Correct Option is A

To find the value of \( x \) in the sample data so that the point estimate of the population mean is 23, we start by calculating the sample mean. The formula for the sample mean (\( \bar{x} \)) is: 

\[ \bar{x} = \frac{\text{Sum of the sample data}}{\text{Number of data points}} \]

Given the sample data is \( 15, 23, x, 37, 19, 32 \) and the number of data points is 6, the mean is:

\[ 23 = \frac{15 + 23 + x + 37 + 19 + 32}{6} \]

Simplify the equation:

\[ 23 = \frac{126 + x}{6} \]

To solve for \( x \), first multiply both sides by 6:

\[ 23 \times 6 = 126 + x \]

Further simplification gives:

\[ 138 = 126 + x \]

Subtract 126 from both sides:

\[ x = 138 - 126 \]

Simplifying gives:

\[ x = 12 \]

Therefore, the value of \( x \) is 12.

Answer 2

The sample mean is given by:

\[ \bar{x} = \frac{\text{Sum of all data points}}{\text{Number of data points}}. \]

Step 1: Substitute the given mean:

We are given that the mean is 23, so substitute it into the formula: \[ 23 = \frac{15 + 23 + x + 37 + 19 + 32}{6}. \]

Step 2: Simplify the equation:

Multiply both sides by 6 to eliminate the denominator: \[ 23 \times 6 = 15 + 23 + x + 37 + 19 + 32. \] This simplifies to: \[ 138 = 126 + x. \]

Step 3: Solve for \( x \):

Subtract 126 from both sides: \[ x = 138 - 126 = 12. \]

Conclusion: Thus, the value of \( x \) is \( 12 \).