The amount of Andy’s monthly phone bill is normally distributed with a mean of RM50 and a standard deviation of RM6. Approximately what percentage of his phone bills are between RM32 and RM68?
Question
The amount of Andy’s monthly phone bill is normally distributed with a mean of RM50 and a standard deviation of RM6. Approximately what percentage of his phone bills are between RM32 and RM68?
Solution
To answer this question, we need to use the properties of the normal distribution.
Step 1: Calculate the Z-scores for RM32 and RM68. The Z-score is a measure of how many standard deviations an element is from the mean.
The formula for the Z-score is:
Z = (X - μ) / σ
where: X is the value we are interested in, μ is the mean, and σ is the standard deviation.
For RM32: Z = (32 - 50) / 6 = -3
For RM68: Z = (68 - 50) / 6 = 3
Step 2: Look up these Z-scores in the Z-table to find the percentage of values to the left of our scores.
The Z-table tells us that the percentage of values to the left of Z = -3 is 0.0013 or 0.13%, and the percentage of values to the left of Z = 3 is 0.9987 or 99.87%.
Step 3: Subtract the percentage of values to the left of Z = -3 from the percentage of values to the left of Z = 3 to find the percentage of values between RM32 and RM68.
0.9987 - 0.0013 = 0.9974 or 99.74%
So, approximately 99.74% of Andy’s phone bills are between RM32 and RM68.
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