The average electric bill in a residential area is $72 for the month of April. The standard deviation is $5. If the amounts of the electric bills are normally distributed, find the probability that the mean of the bill is less than $78.
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To find the probability that the mean of the bill is less than $78, we need to calculate the z-score and then use a normal distribution table or a calculator with a cumulative distribution function.
Step 1: Calculate the z-score
The z-score measures how many standard deviations a data point is from the mean. In this case, we want to find the z-score for $78 using the formula:
z = (x - μ) / σ
where x is the value we are interested in ($78), μ is the mean ($72), and σ is the standard deviation ($5).
z = ($78 - $72) / $5
z = 6 / $5
z = 1.2
Step 2: Use the normal distribution table or calculator
Using a normal distribution table or calculator, we can find the probability associated with the z-score of 1.2. Since we want the probability that the mean of the bill is less than $78, we will look for the area to the left of 1.2.
Using a calculator or software, the probability can be directly obtained by using the cumulative distribution function (CDF). For a z-score of 1.2, the probability is approximately 0.8849.
Therefore, the probability that the mean of the bill is less than $78 is approximately 0.8849, or 88.49% (rounded to two decimal places).