On the seventh day of Statistics, my teacher gave to me --- seven swans a-swimming. But can they swim faster than American Olympic gold medallist Michael Phelps? Michael can swim the 200 meter butterfly in 1 minute 54 seconds (which is 114 seconds). In the same event, the swim times of the population of swans is normally distributed with a mean time of 125 seconds with a standard deviation of 6. What is the probability that a randomly chosen swan would beat Michael’s Olympic record by 1 second or more? (You might wonder why the swans are so slow. Have you ever seen a swan doing the 200 meter butterfly?)
Z = (score-mean)/SD = ([114-1]-125)/6
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.
To find the probability that a randomly chosen swan would beat Michael Phelps' Olympic record by 1 second or more, we need to calculate the area under the normal distribution curve to the right of 1 second.
First, let's standardize Michael Phelps' swim time using the formula: z = (x - μ) / σ, where x is the swim time, μ is the mean, and σ is the standard deviation.
For Michael Phelps:
z = (114 - 125) / 6
z ≈ -1.83
Now, we need to find the probability associated with a z-score of -1.83 using a standard normal distribution table or a statistical tool like Excel, Python, or an online calculator.
The probability associated with a z-score of -1.83 is the area to the left of -1.83 on the standard normal distribution curve. We want to find the area to the right of -1.83, so we subtract the probability we found from 1.
Let's say the probability of a z-score of -1.83 is P(Z ≤ -1.83). The probability that a randomly chosen swan would beat Michael Phelps' Olympic record by 1 second or more is given by:
P(swim time > 1 second faster than Michael Phelps) = 1 - P(Z ≤ -1.83)
You can use a standard normal distribution table or a statistical tool to find the precise value of P(Z ≤ -1.83) and calculate the final probability.