To solve the problems, we will use the properties of the normal distribution. The TI-84 PLUS calculator has functions to help compute probabilities under a normal distribution.
Given:
- Mean (\( \mu \)) = 117 inches
- Standard deviation (\( \sigma \)) = 14 inches
(a) Proportion of trees more than 129 inches tall
We need to calculate \( P(X > 129) \).
-
First, convert 129 inches to a z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \] \[ z = \frac{129 - 117}{14} = \frac{12}{14} \approx 0.8571 \]
-
Then use the calculator:
- Hit the
2nd
button and thenVARS
to access theDISTR
menu. - Select
normalcdf
. - The input will be: \( \textnormal{normalcdf}(129, 1E99, 117, 14) \).
- This calculates the area to the right of 129 inches (meaning more than 129 inches).
- Hit the
Using the calculator, you will approximate this value.
(b) Proportion of trees less than 100 inches tall
We need to calculate \( P(X < 100) \).
-
Convert 100 inches to a z-score: \[ z = \frac{100 - 117}{14} = \frac{-17}{14} \approx -1.2143 \]
-
Use the calculator:
- Again, access the
DISTR
menu. - Select
normalcdf
. - The input will be: \( \textnormal{normalcdf}(-1E99, 100, 117, 14) \).
- Again, access the
(c) Probability that a randomly chosen tree is between 85 and 100 inches tall
We need to calculate \( P(85 < X < 100) \).
-
Convert both values to z-scores. For 85: \[ z = \frac{85 - 117}{14} = \frac{-32}{14} \approx -2.2857 \]
For 100 (already done): \[ z \approx -1.2143 \]
-
Use the calculator:
- Access the
DISTR
menu again. - Select
normalcdf
. - The input will be: \( \textnormal{normalcdf}(85, 100, 117, 14) \).
- Access the
Summary of Steps on TI-84:
- Use
normalcdf(lower, upper, μ, σ)
where:- For \( P(X > 129) \):
normalcdf(129, 1E99, 117, 14)
- For \( P(X < 100) \):
normalcdf(-1E99, 100, 117, 14)
- For \( P(85 < X < 100) \):
normalcdf(85, 100, 117, 14)
- For \( P(X > 129) \):
Final Calculations
To complete the calculations and get numeric answers, you would enter these expressions into the TI-84 calculator to evaluate the probabilities.
Without a calculator to compute the exact decimals, you would be able to obtain those four decimal values directly upon entering the appropriate commands for each scenario.