length of time in minutes required to complete exercises is distributed with a mean of 60 minutes and a std. deviation of 8 minutes. what percent will require between 60 and 76 minutes to complete the exercises? what percent of participants will require more than 72 minutes to complete th exercises? what percent will require between 50 and 72 minutes to complete the exercises?
Use z-scores and a z-table to find your percentages.
Let's look at the very last question and then see if you can figure out the other two questions.
Find two z-scores using this formula:
z = (x - mean)/sd
mean = 60
sd = 8
x = 50, 72
Therefore:
z = (50 - 60)/8 = ?
z = (72 - 60)/8 = ?
I'll let you finish the calculations.
Once you have the z-scores, check a z-table for the probability between those two scores. Once you have that value, convert to a percent.
I hope this will help get you started.
To solve these questions, we will use the properties of the standard normal distribution, also known as the Z-distribution.
The Z-distribution is a mathematical model that represents a normal distribution with a mean of 0 and a standard deviation of 1. We can use this distribution to standardize any normal distribution by calculating the Z-score.
The Z-score (Z) is calculated using the formula:
Z = (X - μ) / σ
Where,
X is the value we are interested in,
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
Now let's solve the questions one by one:
1. What percent will require between 60 and 76 minutes to complete the exercises?
To find this percentage, we need to find the area under the normal curve between these two values.
First, let's calculate the Z-score for 60 minutes:
Z1 = (60 - 60) / 8 = 0
Next, we calculate the Z-score for 76 minutes:
Z2 = (76 - 60) / 8 = 2
Now, we need to find the area between these two Z-scores, which represents the percentage of the distribution. We can use the Z-table or a statistical calculator to find the area between these two Z-scores.
Using a Z-table, we find that the area to the left of Z = 2 is approximately 0.9772, and the area to the left of Z = 0 is 0.5. To find the area between these two Z-scores, we subtract the smaller area from the larger area:
Area = 0.9772 - 0.5 = 0.4772
Therefore, the percentage of participants requiring between 60 and 76 minutes is approximately 47.72%.
2. What percent of participants will require more than 72 minutes to complete the exercises?
To find this percentage, we need to calculate the area to the right of 72 minutes.
First, we calculate the Z-score for 72 minutes:
Z = (72 - 60) / 8 = 1.5
Using the Z-table, we find that the area to the left of Z = 1.5 is approximately 0.9332.
To find the area to the right of Z = 1.5, we subtract the area to the left from 1:
Area = 1 - 0.9332 = 0.0668
Therefore, the percentage of participants requiring more than 72 minutes is approximately 6.68%.
3. What percent will require between 50 and 72 minutes to complete the exercises?
To find this percentage, we perform similar steps as in the first question.
First, we calculate the Z-score for 50 minutes:
Z1 = (50 - 60) / 8 = -1.25
Next, the Z-score for 72 minutes:
Z2 = (72 - 60) / 8 = 1.5
Using the Z-table, we find that the area to the left of Z = -1.25 is approximately 0.1056, and the area to the left of Z = 1.5 is approximately 0.9332. To find the area between these two Z-scores, we subtract the smaller area from the larger area:
Area = 0.9332 - 0.1056 = 0.8276
Therefore, the percentage of participants requiring between 50 and 72 minutes is approximately 82.76%.