The mean of a normal probability distribution is 440; the standard deviation is 8.
A: About 95% of the observations lie between what two values?
B: Practically all of the observations lie between what to values?
A: 2 s.d. below and 2 s.d. above the mean
... [440 - (2 * 8)] and [440 + (2 * 8)]
B: 3 s.d. below and 3 s.d. above the mean
... [440 - (3 * 8)] and [440 + (3 * 8)]
A: To find the values between which about 95% of the observations lie, we can use the rule of thumb for normal distributions, also known as the 68-95-99.7 rule. According to this rule, approximately 95% of the observations lie within two standard deviations (σ) of the mean (μ) in a normal distribution.
So, for this question, we can calculate the values by adding and subtracting two standard deviations from the mean.
The lower value would be: 440 - (2 x 8) = 440 - 16 = 424.
The upper value would be: 440 + (2 x 8) = 440 + 16 = 456.
Therefore, about 95% of the observations lie between 424 and 456.
B: Practically all of the observations lie within three standard deviations (σ) of the mean (μ) in a normal distribution. So, to find the values between which practically all of the observations lie, we can calculate by adding and subtracting three standard deviations from the mean.
The lower value would be: 440 - (3 x 8) = 440 - 24 = 416.
The upper value would be: 440 + (3 x 8) = 440 + 24 = 464.
Therefore, practically all of the observations lie between 416 and 464.
To find the values between which a certain percentage of observations lie in a normal probability distribution, we can use the concept of standard deviations and the Empirical Rule (also known as the 68-95-99.7 Rule). This rule states that:
- Approximately 68% of the observations lie within one standard deviation of the mean.
- Approximately 95% of the observations lie within two standard deviations of the mean.
- Approximately 99.7% of the observations lie within three standard deviations of the mean.
Given that the mean of the normal distribution is 440 and the standard deviation is 8, we can calculate the values for the given percentages:
A: To find the range between which about 95% of the observations lie, we need to consider two standard deviations.
1. Calculate the lower value:
Lower value = Mean - (2 * Standard deviation)
Lower value = 440 - (2 * 8) = 424
2. Calculate the upper value:
Upper value = Mean + (2 * Standard deviation)
Upper value = 440 + (2 * 8) = 456
Therefore, about 95% of the observations lie between the values 424 and 456.
B: To find the range between which practically all of the observations lie, we need to consider three standard deviations.
1. Calculate the lower value:
Lower value = Mean - (3 * Standard deviation)
Lower value = 440 - (3 * 8) = 416
2. Calculate the upper value:
Upper value = Mean + (3 * Standard deviation)
Upper value = 440 + (3 * 8) = 464
Therefore, practically all of the observations lie between the values 416 and 464.