Let x denote the time taken to run a road race. Suppose x is approximately normally distributed with a mean of 190 minutes and a standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race in 225 to 235 minutes?
To find the probability that a runner will complete the road race in the time range of 225 to 235 minutes, we need to calculate the probability within that range using the given mean and standard deviation of the normal distribution.
First, we need to standardize the values of 225 and 235 using the formula:
z = (x - μ) / σ
where z is the standard score, x is the given value, μ is the mean, and σ is the standard deviation.
For 225 minutes:
z1 = (225 - 190) / 21
For 235 minutes:
z2 = (235 - 190) / 21
Next, we find the cumulative probability for each standardized value using a standard normal distribution table or a calculator. The cumulative probability represents the area under the normal curve to the left of the standardized value.
P(z < z1) = P(z < (225 - 190) / 21)
P(z < z1) = P(z < 1.6667)
P(z < z2) = P(z < (235 - 190) / 21)
P(z < z2) = P(z < 2.1429)
Finally, we calculate the desired probability by subtracting the cumulative probability for z1 from the cumulative probability for z2:
P(225 < x < 235) ≈ P(z1 < z < z2) = P(z < z2) - P(z < z1)
Now you can use a standard normal distribution table or a calculator to find the probabilities associated with z1 and z2 and subtract them to get the final probability.