Use the TI-84 calculator to find the Z-scores that bound the middle 82% of the area under the standard normal curve.

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Use the TI-84 calculator to find the Z-scores that bound the middle 82% of the area under the standard normal curve. Enter the answers in ascending order and round to two decimal places. The Z-scores for the given area are [insert correct values here] and [insert correct values here].

The Correct Answer and Explanation is:

To find the Z-scores that bound the middle 82% of the area under the standard normal curve using the TI-84 calculator, follow these steps:

Steps to Calculate:

  1. Press the 2nd button, then the VARS button to access the DISTR menu (where probability distributions are listed).
  2. Scroll down and select invNorm(. This function will allow you to find the Z-scores for a given probability.
  3. The syntax for the invNorm function is: invNorm(area to the left, mean, standard deviation).
    • In this case, the mean is 0 and the standard deviation is 1 (because we are dealing with the standard normal distribution).
  4. Middle 82% of the area means that 82% of the area is between the two Z-scores, and 18% is split equally on both sides of the middle. This means:
    • The area to the left of the lower Z-score is 9% (half of 18%).
    • The area to the right of the upper Z-score is also 9%.

Thus, the area to the left of the lower Z-score is 0.09, and the area to the left of the upper Z-score is 0.91 (since the total area under the curve is 1, and 1 – 0.09 = 0.91).

  1. First Z-score (lower bound):
    • For the lower Z-score, enter invNorm(0.09, 0, 1) into the calculator.
    • The result is approximately -1.34.
  2. Second Z-score (upper bound):
    • For the upper Z-score, enter invNorm(0.91, 0, 1) into the calculator.
    • The result is approximately 1.34.

Final Answer:

The Z-scores that bound the middle 82% of the area under the standard normal curve are -1.34 and 1.34.

Explanation:

  • A Z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution.
  • The middle 82% means that 9% of the area is on the lower tail, and 9% on the upper tail, leaving 82% in the middle. By using the invNorm function, we are finding the Z-scores that correspond to the cumulative probabilities of 0.09 (lower bound) and 0.91 (upper bound).
  • The TI-84 calculator uses the cumulative distribution function (CDF) to return these Z-scores.

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