A bookstore sells books for $2, $3, $5, and $10. Let random variable X = "amount of money for one book."Look at the relative-frequency table below representing the amount of money spent on one item and the relative frequencies with which customers purchase them:x P(X = x)2 .553 .265 .1110 .08If the expected amount of money spent by a customer is $3.23, what is the standard deviation?
Question
A bookstore sells books for 3, 10. Let random variable X = "amount of money for one book."Look at the relative-frequency table below representing the amount of money spent on one item and the relative frequencies with which customers purchase them:x P(X = x)2 .553 .265 .1110 .08If the expected amount of money spent by a customer is $3.23, what is the standard deviation?
Solution
To calculate the standard deviation, we first need to calculate the variance. The variance is the average of the squared differences from the mean. Here are the steps to calculate the variance:
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Calculate the squared difference from the mean for each value of X. This is done by subtracting the mean from each value of X, squaring the result, and then multiplying by the probability of that value of X.
For X = 2: (2 - 3.23)^2 * 0.55 = 0.7085 For X = 3: (3 - 3.23)^2 * 0.26 = 0.0156 For X = 5: (5 - 3.23)^2 * 0.11 = 0.2161 For X = 10: (10 - 3.23)^2 * 0.08 = 3.4624
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Sum up these values to get the variance.
Variance = 0.7085 + 0.0156 + 0.2161 + 3.4624 = 4.4026
The standard deviation is the square root of the variance. So, the standard deviation is sqrt(4.4026) = 2.10 (rounded to two decimal places).
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