A rescue squad spokesperson claims that the squad receives an average of 3 calls per hour. If at most 6 calls were received over a 5-hour period, would you suspect that the spokesperson was overestimating the

average number of calls received per hour? Explain based on the probability
of this occurring.

If the expected number is 3/hr then in 5hrs there should be 5*3=15calls. It would not be real unusual to have 6 calls, the answer really depends on what the variance is, not just the average. If the variance is large this could be a perfectly acceptable number.

To determine if the spokesperson was overestimating the average number of calls received per hour, we need to consider the probability of receiving at most 6 calls over a 5-hour period given an average of 3 calls per hour.

Based on the information provided, the expected number of calls in a 5-hour period would be 5 hours * 3 calls per hour = 15 calls.

To calculate the probability of receiving at most 6 calls in a 5-hour period, we can use a probability distribution such as the Poisson distribution, assuming the calls occur randomly and independently with a constant average rate.

Using the Poisson distribution formula, the probability of receiving exactly k calls in a given time period is given by the formula:

P(k) = (e^(-λ) * λ^k) / k!

where λ is the average number of calls per time period.

In this case, λ = 3 calls per hour * 5 hours = 15.

Now, let's calculate the probability of receiving at most 6 calls in a 5-hour period:

P(≤6 calls) = P(k=0) + P(k=1) + P(k=2) + P(k=3) + P(k=4) + P(k=5) + P(k=6)

Using the Poisson distribution formula, we can substitute the values into the formula for each value of k and sum them up:

P(≤6 calls) = (e^(-15) * 15^0) / 0! + (e^(-15) * 15^1) / 1! + (e^(-15) * 15^2) / 2! + (e^(-15) * 15^3) / 3! + (e^(-15) * 15^4) / 4! + (e^(-15) * 15^5) / 5! + (e^(-15) * 15^6) / 6!

Using a calculator or statistical software, we can find the numerical value of this probability.

If the probability of receiving at most 6 calls in a 5-hour period is relatively high (close to 1), then it would not be surprising or unusual to have 6 calls. However, if the probability is low (close to 0), it would suggest that the spokesperson's average estimate of 3 calls per hour may be too high.

It is important to note that in this case, we have only considered the probability and not the variance. If the variance is large, there can be more fluctuations around the average, making 6 calls per 5-hour period acceptable.

To make a conclusive judgment, additional information about the variance of the number of calls would be needed.