Suppose an automobile association reports that the average time it takes to respond to an emergency call is 25 minutes. Assume the variable is approximately normally distributed and the standard deviation is 4.5 minutes. If 104 calls are randomly selected, approximately how many will be responded to in less than 22 minutes?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z score. Multiply that by 104.
To solve this problem, we'll use the z-score formula to find the proportion of calls that will be responded to in less than 22 minutes.
The formula for the z-score is:
z = (x - μ) / σ
where:
- x is the value we're interested in (in this case, 22 minutes)
- μ is the mean (average) response time (25 minutes)
- σ is the standard deviation (4.5 minutes)
Calculating the z-score:
z = (22 - 25) / 4.5
= -3 / 4.5
= -0.67
Looking up the z-score in the normal distribution table, we find that the proportion of calls responded to in less than 22 minutes is approximately 0.2514.
Therefore, if 104 calls are randomly selected, the approximate number of calls responded to in less than 22 minutes would be:
104 * 0.2514 = 26.17
Rounding to the nearest whole number, approximately 26 calls will be responded to in less than 22 minutes.
To find the approximate number of calls that will be responded to in less than 22 minutes, we need to calculate the z-score and use it to look up the corresponding probability in the standard normal distribution table.
The z-score formula is given by:
z = (X - μ) / σ
Where:
X = 22 minutes (desired value)
μ = 25 minutes (mean)
σ = 4.5 minutes (standard deviation)
Substituting the values into the formula:
z = (22 - 25) / 4.5
z ≈ -0.67
Now, we need to find the probability of getting a value less than -0.67 in the standard normal distribution table.
Using the z-score table or a statistical calculator, we find that the area to the left of -0.67 is approximately 0.2514.
Therefore, approximately 25.14% of the calls will be responded to in less than 22 minutes.
To find the number of calls out of 104, we multiply the probability by the total number of calls:
Number of calls = Probability * Total number of calls
Number of calls = 0.2514 * 104
Number of calls ≈ 26
Therefore, approximately 26 calls will be responded to in less than 22 minutes out of a sample of 104 calls.