The height of seaweed of all plants in a body of water are normally distributed with a mean of $10 \mathrm{~cm}$ and a standard deviation of $2 \mathrm{~cm}$. Which length separates the lowest $30 \%$ of the means of the plant heights in a sampling distribution of sample size 15 from the highest $70 \%$ ? Round your answers to the nearest hundredth. Use the z-table below: \begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|} \hline$z$ & 0.00 & 0.01 & 0.02 & 0.03 & 0.04 & 0.05 & 0.06 & 0.07 & 0.08 & 0.09 \\ \hline \hline-0.8 & 0.212 & 0.209 & 0.206 & 0.203 & 0.201 & 0.198 & 0.195 & 0.192 & 0.189 & 0.187 \\ \hline-0.7 & 0.242 & 0.239 & 0.236 & 0.233 & 0.230 & 0.227 & 0.224 & 0.221 & 0.218 & 0.215 \\ \hline-0.6 & 0.274 & 0.271 & 0.268 & 0.264 & 0.261 & 0.258 & 0.255 & 0.251 & 0.248 & 0.245 \\ \hline-0.5 & 0.309 & 0.305 & 0.302 & 0.298 & 0.295 & 0.291 & 0.288 & 0.284 & 0.281 & 0.278 \\ \hline-0.4 & 0.345 & 0.341 & 0.337 & 0.334 & 0.330 & 0.326 & 0.323 & 0.319 & 0.316 & 0.312 \\ \hline-0.3 & 0.382 & 0.378 & 0.374 & 0.371 & 0.367 & 0.363 & 0.359 & 0.356 & 0.352 & 0.348 \\ \hline \hline \end{tabular} Round the $z$-score and $\bar{x}$ to two decimal places. Provide your answer below: \[ \begin{array}{l} z \text {-score }=\square \\ \bar{x}=\square \mathrm{cm} \end{array} \]
Solution
Step 1 ：First, we need to find the z-score for the 30th percentile. We can use the provided z-table for this. The z-table gives the cumulative probability (the area to the left under the curve) for each z-score. We need to find the z-score that corresponds to a cumulative probability of 0.30.
Step 2 ：The z-score for the 30th percentile is approximately -0.52 when rounded to two decimal places.
Step 3 ：Now, let's use this z-score to find the corresponding height (x̄) using the formula for a z-score: \(x̄ = z*(σ/√n) + μ\).
Step 4 ：Substitute the z-score, the given values for μ, σ, and n into the formula: \(x̄ = -0.52*(2/√15) + 10\).
Step 5 ：The mean of the plant heights that separates the lowest 30% from the highest 70% is approximately 9.73 cm when rounded to two decimal places.
Step 6 ：Final Answer: The z-score is \(\boxed{-0.52}\) and the mean of the plant heights is \(\boxed{9.73 \mathrm{~cm}}\).
