To determine how many text messages separate the lowest 15% from the highest 85% in a sampling distribution of 144 teenagers, we need to follow these steps carefully:
1. Identify the given parameters:
- Population mean (μ): 312
- Population standard deviation (σ): 125
- Sample size (n): 144
- Percentile to find the threshold for: 15%
2. Find the corresponding z-score for the 15th percentile using the z-table.
Looking up the z-table, we see that the z-score for the 15th percentile is approximately -1.04.
3. Calculate the standard deviation of the sampling distribution (standard error):
[tex]\[
\text{Standard Error} = \frac{\sigma}{\sqrt{n}} = \frac{125}{\sqrt{144}} = \frac{125}{12} \approx 10.42
\][/tex]
4. Calculate the value (x) that separates the lowest 15% using the z-score formula:
[tex]\[
x = \mu + z \cdot (\text{Standard Error})
\][/tex]
Replacing the known values we get:
[tex]\[
x = 312 + (-1.04) \cdot 10.42 \approx 312 - 10.83 \approx 301
\][/tex]
Therefore, the z-score and the corresponding number of text messages that separate the lowest 15% are:
[tex]\[
\begin{array}{l}
z \text {-score }=-1.04 \\
\bar{x}=301 \text { text messages }
\end{array}
\][/tex]
