Question

Consider normally distributed data with a mean of 35 and a standard deviation of 2. If there are 350 values between 31 and 33, how many values are there in the distribution?

Answer 1

Hi,

7955.

Step-by-step explanation:

`Z=(x-\mu)/(\sigma) =(x-35)/(2)\\\\Z_1=(31-35)/(2)=-2\\Z_2=(32-35)/(2)=-1.5\\p(31 \leq x \leq 32)\\=p(-2 \leq Z \leq -1.5)\\=p(1.5 \leq Z \leq 2)\\ = 0.9772-0.9332\\=0.044\\=4.4 \%\\350*100\%/4.4\%=7954,54 \approx {7955}`

Answer 2

To find the total number of values in the distribution, we use the z-score formula to find the proportion of values between 31 and 33. We then multiply this proportion by the total number of values given to solve for the unknown value.

Step-by-step explanation:

The mean of normally distributed data is 35 and the standard deviation is 2. We are given that there are 350 values between 31 and 33. To find the total number of values in the distribution, we can use the z-score formula.

First, we find the z-scores for 31 and 33 using the formula:
z = (x - mean) / standard deviation
z for 31 = (31 - 35) / 2 = -2
z for 33 = (33 - 35) / 2 = -1

The z-scores represent the number of standard deviations each value is from the mean. We can then find the corresponding area under the curve for these z-scores using a standard normal distribution table or a calculator.

The area between z = -1 and z = -2 represents the proportion of values between 31 and 33. We can subtract this area from 0.5 (which represents 50% of the values on each side of the mean) to find the proportion of values below 31.
Area between z = -1 and z = -2 = 0.1359
Proportion of values below 31 = 0.5 - 0.1359 = 0.3641

To find the actual number of values below 31, we multiply this proportion by the total number of values in the distribution:
Number of values below 31 = 0.3641 * (total number of values) = 0.3641 * (350 + unknown)

We can now solve for the unknown total number of values:
0.3641 * (350 + unknown) = 350
0.3641 * unknown = 0.3641 * 350
unknown = 350 - 0.3641 * 350

Therefore, the total number of values in the distribution is the unknown value calculated above.