Suppose you roll a fair, 4-sided die once every day for 360 days.
[Image source: https://commons.wikimedia.org/wiki/File:4-sided_dice_250.jpg (Links to an external site.) ; Author: Fantasy; CC BY-SA 3.0 (Links to an external site.)]
Let X represent the number of rolls that land with "1" at the bottom edge of the die, as in the illustration above. Note: the probability of this event is 0.25 in any single roll.
X is a binomial random variable. What are the parameters n and p for its distribution?
Without evaluating the expression, use the binomial formula to write out exactly P(X = 81). You should not evaluate the binomial coefficient, nor the exponential expressions. Do not give a decimal approximation; use Formula 5.1 from our textbook (Section 5.3).
We want to use the normal approximation to the binomial to approximate the probability P(85 < X ≤ 95).
Check that normal approximation is reasonable in this case. (This is Step 2 of Procedure 6.3, p. 299 in our textbook.)
What are the mean and standard deviation of the appropriate normal distribution? (Step 3 of Procedure 6.3.)
What is the correction for continuity to evaluate this probability? I.e., what are the endpoints a and b so that, if Y denotes the normal random variable, we have P(85 < X ≤ 95) ≈ P(a ≤ Y ≤ b) ?
Find the probability given by the approximation. If possible, also find the exact value of P(85 < X ≤ 95), using technology. How does the approximation compare to the exact value?
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Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
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