As reported in Runner's World magazine, the times of the finishers in the new york 10 km run are normally distrubted with a mean of 1.02 hours and a standard deviation of 9 minutes. Let X denote finishing time for the finishers. Then,
A. Find the percentage of finishers with times between 50 and 70 minutes
B. find the percentage of finishers with times between 60 and 75 minutes.
How would I approach the first one. This is what I did for half of the problem
50-1.02/9= 49.9
I don't know where to go from there
You have to find the z-score for both 50 and 70 minutes. Change the 1.02 hours to 61.2 minutes for the mean.
z = (50 -61.2)/9
do the same for the 70 minutes.
go to a standard z-table and find the area between those two values.
Do the same thing for B.
Your mistake was not having the same units. Use minutes with minutes and not hours with minutes.
New York City 10-km Run. As reported in Runner’s World magazine, the times of the finishers in the New York City 10-km run are normally distributed with mean 61 minutes and standard deviation 9 minutes. Let x denote finishing time for finishers in this race.
a. Sketch the distribution of the variable x.
b. Obtain the standardized version, z, of x.
c. Identify and sketch the distribution of z.
d. The percentage of finishers with times between 50 and 70 minutes is equal to the area under the standard normal curve between _______ and _______.
e. The percentage of finishers with times less than 75 minutes is equal to the area under the standard normal curve that lies to the _______ of _______.
To find the percentage of finishers with times between 50 and 70 minutes, you need to calculate the z-scores for the lower and upper bounds and then use a standard normal distribution table or calculator.
The formula to calculate the z-score is:
z = (x - μ) / σ
where:
x = value being standardized (in this case, the finishing time)
μ = mean of the distribution
σ = standard deviation of the distribution
For the lower bound of 50 minutes:
z_lower = (50 - 60) / 9
For the upper bound of 70 minutes:
z_upper = (70 - 60) / 9
Once you have the z-scores, you can refer to a standard normal distribution table or use a calculator that provides the area under the standard normal curve to find the corresponding percentage.
Let's calculate the z-scores:
z_lower = (50 - 60) / 9 = -1.11
z_upper = (70 - 60) / 9 = 1.11
Now, using the standard normal distribution table or calculator, you can find the corresponding percentages for z-scores of -1.11 and 1.11. Subtracting the lower percentage value from the higher percentage value will give you the percentage of finishers with times between 50 and 70 minutes.
To find the percentage of finishers with times between 50 and 70 minutes, you need to convert these times into standard units using the given mean and standard deviation.
First, let's convert the lower and upper bounds of the range into standard units:
Lower bound: (50 - 60) minutes = -10 minutes
Upper bound: (70 - 60) minutes = 10 minutes
Next, divide these standard units by the standard deviation:
z-score for lower bound = (-10 minutes) / (9 minutes) = -1.11 (approximately)
z-score for upper bound = (10 minutes) / (9 minutes) = 1.11 (approximately)
Now that we have the z-scores, we can find the area under the normal distribution curve between these two z-scores. This will give us the percentage of finishers with times between 50 and 70 minutes.
You can use a standard normal distribution table or a calculator with a cumulative normal distribution function to find this area.
Using a standard normal distribution table, you would lookup the z-values (-1.11 and 1.11) and find the corresponding areas. Then, subtract the smaller area from the larger area to get the percentage of finishers between 50 and 70 minutes.
If you use a calculator with a cumulative normal distribution function, you can input the lower and upper z-scores to get the percentage directly.
Keep in mind that the cumulative normal distribution function gives the area to the left of the given z-score. So, you may need to calculate the area for the lower z-score and subtract it from 1 to get the area between the two z-scores.
Therefore, the steps to calculate the percentage of finishers with times between 50 and 70 minutes are:
1. Convert the time range into standard units using the mean and standard deviation.
2. Calculate the z-scores for the lower and upper bounds.
3. Use a standard normal distribution table or a calculator with a cumulative normal distribution function to find the areas corresponding to these z-scores.
4. Subtract the smaller area from the larger area to get the percentage of finishers between 50 and 70 minutes.