Given the length an athlete throws a hammer is a normal random variable with mean 50 feet and standard deviation 5 feet, what is the probability he throws it between 50 feet and 60 feet?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion between the two Z scores.
0.5
To calculate the probability that the athlete throws the hammer between 50 feet and 60 feet, we need to standardize the values and then use the standard normal distribution.
Let's denote the random variable X as the length the athlete throws the hammer.
First, we need to standardize the values of 50 feet and 60 feet. To do this, we can use the formula for standardization:
Z = (X - μ) / σ
Where:
X is the observed value (in this case, the length of the throw),
μ is the mean of the distribution, which is 50 feet, and
σ is the standard deviation of the distribution, which is 5 feet.
For the lower bound of 50 feet:
Z1 = (50 - 50) / 5 = 0
For the upper bound of 60 feet:
Z2 = (60 - 50) / 5 = 2
Now, we will use the standard normal distribution table or a calculator to find the probabilities associated with these z-scores.
The probability that the athlete throws the hammer between 50 feet and 60 feet can be calculated as the difference between the cumulative probabilities corresponding to the upper and lower z-scores:
P(50 < X < 60) = P(0 < Z < 2)
Using the standard normal distribution table or calculator, we find:
P(0 < Z < 2) ≈ 0.4772
Therefore, the probability that the athlete throws the hammer between 50 feet and 60 feet is approximately 0.4772, or 47.72%.
To find the probability that the athlete throws the hammer between 50 feet and 60 feet, we need to calculate the cumulative probability up to 60 feet and subtract the cumulative probability up to 50 feet.
To calculate the cumulative probability, we'll use the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Since we're given the mean and standard deviation for the hammer throw, we'll need to standardize the values.
To standardize a value, we subtract the mean and then divide by the standard deviation. In this case, for 50 feet:
Z = (X - mean) / standard deviation
Z = (50 - 50) / 5
Z = 0
So, for 50 feet, the standard score (Z-score) is 0.
For 60 feet:
Z = (X - mean) / standard deviation
Z = (60 - 50) / 5
Z = 2
The standard score for 60 feet is 2.
Next, we can use a standard normal distribution table or a statistical software to find the corresponding cumulative probabilities for these standard scores. The cumulative probability at a standard score of 0 is 0.5000, and the cumulative probability at a standard score of 2 is 0.9772.
To find the probability of the athlete throwing the hammer between 50 feet and 60 feet, we subtract the cumulative probability at 50 feet from the cumulative probability at 60 feet:
P(50 ≤ X ≤ 60) = P(X ≤ 60) - P(X ≤ 50)
= 0.9772 - 0.5000
= 0.4772
Therefore, the probability that the athlete throws the hammer between 50 feet and 60 feet is 0.4772 or 47.72%.