A survey was conducted to determine the number of hours people listen to the radio. While most of the listening is in the car, it turns out that the mean is 2.5 hours with a standard deviation of 0.75 hours. Find the probability that the sum of 100 values is less than 240.
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To find the probability that the sum of 100 values is less than 240, we first need to calculate the mean and standard deviation of the sum.
The mean of the sum is the sum of the means of the individual values, which is given by:
Mean_sum = Mean_individual * Number_of_values = 2.5 * 100 = 250
The standard deviation of the sum is the square root of the sum of the variances of the individual values, which is given by:
SD_sum = sqrt(Variance_individual * Number_of_values) = sqrt((0.75)^2 * 100) = sqrt(56.25) = 7.5
To find the probability that the sum of 100 values is less than 240, we can use the standard normal distribution. We need to calculate the z-score for the value 240 using the formula:
z = (X - Mean_sum) / SD_sum
where X is the value of interest.
In this case, X = 240, Mean_sum = 250, and SD_sum = 7.5. Hence, the z-score is:
z = (240 - 250) / 7.5 = -10 / 7.5 = -1.33 (rounded to two decimal places)
Using a standard normal distribution table or a calculator, we can find the cumulative probability for a z-score of -1.33. The probability is the area to the left of the z-score.
Looking up the z-score in a standard normal distribution table, we find that the cumulative probability for a z-score of -1.33 is approximately 0.0918.
Therefore, the probability that the sum of 100 values is less than 240 is approximately 0.0918 or 9.18%.