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.
To determine the value of \( x \) in the provided sample data such that the population mean's point estimate is 23, we first compute the sample mean. The sample mean (\( \bar{x} \)) is calculated using the formula:
\[ \bar{x} = \frac{\text{Sum of the sample data}}{\text{Number of data points}} \]
Given the sample data points \( 15, 23, x, 37, 19, 32 \) and a total of 6 data points, the mean equation is:
\[ 23 = \frac{15 + 23 + x + 37 + 19 + 32}{6} \]
This equation simplifies to:
\[ 23 = \frac{126 + x}{6} \]
To solve for \( x \), multiply both sides of the equation by 6:
\[ 23 \times 6 = 126 + x \]
Performing the multiplication yields:
\[ 138 = 126 + x \]
Finally, isolate \( x \) by subtracting 126 from both sides:
\[ x = 138 - 126 \]
The result is:
\[ x = 12 \]
Thus, the value of \( x \) is 12.
| \(\text{Length (in mm)}\) | 70-80 | 80-90 | 90-100 | 100-110 | 110-120 | 120-130 | 130-140 |
|---|---|---|---|---|---|---|---|
| \(\text{Number of leaves}\) | 3 | 5 | 9 | 12 | 5 | 4 | 2 |