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herry trees in a certain orchard have heights that are normally distributed with mean U=117 inches and standard deviation O = 14 inches. Use the TI-84 PLUS calculator to answer the following. Round the answers to at least four decimal places.

(a) What proportion of trees are more than 129 inches tall?

(b) What proportion of trees are less than 100 inches tall?

(c) What is the probability that a randomly chosen tree is between 85 and 100 inches tall?

1 answer

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 \( P(X > 129) \).

  1. First, convert 129 inches to a z-score using the formula: \[ z = \frac{X - \mu}{\sigma} \] \[ z = \frac{129 - 117}{14} = \frac{12}{14} \approx 0.8571 \]

  2. Then use the calculator:

    • Hit the 2nd button and then VARS to access the DISTR 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).

Using the calculator, you will approximate this value.

(b) Proportion of trees less than 100 inches tall

We need to calculate \( P(X < 100) \).

  1. Convert 100 inches to a z-score: \[ z = \frac{100 - 117}{14} = \frac{-17}{14} \approx -1.2143 \]

  2. Use the calculator:

    • Again, access the DISTR menu.
    • Select normalcdf.
    • The input will be: \( \textnormal{normalcdf}(-1E99, 100, 117, 14) \).

(c) Probability that a randomly chosen tree is between 85 and 100 inches tall

We need to calculate \( P(85 < X < 100) \).

  1. Convert both values to z-scores. For 85: \[ z = \frac{85 - 117}{14} = \frac{-32}{14} \approx -2.2857 \]

    For 100 (already done): \[ z \approx -1.2143 \]

  2. Use the calculator:

    • Access the DISTR menu again.
    • Select normalcdf.
    • The input will be: \( \textnormal{normalcdf}(85, 100, 117, 14) \).

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)

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.