Using standard deviation. Normally distributed mean of 0 degrees standard deviation of 1.00 degrees celcius.
1. between 0 and 1.28
2. greater than 0.37
3. less than -0.92
4. between 1.50 and 2.50
5. between -0.90 and 1.95
6. less than 0
I don't know how to solve these problems.
They are asking for the fraction of the distribution that lies within each of the stated intervals. You can solve these with a table lookup of the normal distribution function, which should be explaied in your text, or by using the handy Java computional tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html
For the first problem, enter 1.00 for the standard deviation, 0 for the mean, and integration limits "start" and "end" of 0 and 1.28, and then hit the "enter " key. I get 39.97% for that one. The last one (6) is 50%, since the distribution curve is symmetic and you are taking the integral under the curve beyond the mean value. For (4), I get 6.06%. When you are only given one limit, such as in problem (3), take a large number like -10 (ten standard deviations) as the lower limit, since you cannot enter minus infinity as a limit.
To solve these problems using standard deviation, you can follow these steps:
1. For problem 1, you are looking for the fraction of the distribution that lies between 0 and 1.28 standard deviations above the mean.
- Use a normal distribution table or a computational tool like the one mentioned to calculate the area under the curve between 0 and 1.28.
- Enter the standard deviation as 1.00, the mean as 0, and the integration limits as 0 and 1.28.
- The result should be 39.97%, indicating that approximately 39.97% of the distribution lies between 0 and 1.28 standard deviations above the mean.
2. For problem 2, you want to find the fraction of the distribution that is greater than 0.37 standard deviations above the mean.
- Similar to problem 1, you can use the normal distribution table or the computational tool to calculate the area under the curve from 0.37 standard deviations above the mean to positive infinity.
- Enter the standard deviation as 1.00, the mean as 0, and the integration limits as 0.37 and a large number (to approximate infinity).
- The result will give you the fraction of the distribution greater than 0.37 standard deviations above the mean.
3. For problem 3, you need to determine the fraction of the distribution that is less than -0.92 standard deviations below the mean.
- As mentioned earlier, you can use the normal distribution table or the computational tool to calculate the area under the curve from negative infinity to -0.92 standard deviations below the mean.
- Enter the standard deviation as 1.00, the mean as 0, and the integration limits as a large negative number (approximating negative infinity) and -0.92.
- The result will give you the fraction of the distribution less than -0.92 standard deviations below the mean.
4. For problem 4, you want to find the fraction of the distribution that lies between 1.50 and 2.50 standard deviations above the mean.
- Use the normal distribution table or the computational tool to calculate the area under the curve between 1.50 and 2.50 standard deviations above the mean.
- Enter the standard deviation as 1.00, the mean as 0, and the integration limits as 1.50 and 2.50.
- The result will give you the fraction of the distribution that lies between 1.50 and 2.50 standard deviations above the mean.
5. For problem 5, you need to determine the fraction of the distribution that lies between -0.90 and 1.95 standard deviations above the mean.
- Use the normal distribution table or the computational tool to calculate the area under the curve between -0.90 and 1.95 standard deviations above the mean.
- Enter the standard deviation as 1.00, the mean as 0, and the integration limits as -0.90 and 1.95.
- The result will give you the fraction of the distribution that lies between -0.90 and 1.95 standard deviations above the mean.
6. For problem 6, you are looking for the fraction of the distribution that is less than 0 standard deviations above the mean, which is the left half of the normal distribution curve.
- Since the distribution curve is symmetric and you are taking the integral under the curve beyond the mean value, the result will be 50%.
- There is no need to perform any calculations for this problem.
These steps should help you solve the problems using standard deviation.