Observations are drawn from a bell-shaped distribution with a mean of 40 and a standard deviation of 3. There are 300 observations in the data set.
a. Approximately what percentage of the observations are less than 46? (Round your answer to 1 decimal place.)
b. Approximately how many observations are less than 46? (Round your answer to the nearest whole number.)
a. To determine the percentage of observations that are less than 46, we can use the standard normal distribution.
First, we need to calculate the z-score for 46 using the formula:
z = (x - μ) / σ
where x is the value we are interested in (46), μ is the mean (40), and σ is the standard deviation (3).
z = (46 - 40) / 3
z = 2
Next, we can use a standard normal distribution table or a calculator to find the area under the curve to the left of z = 2.
Looking up the z-score of 2 in a standard normal table, we find that the area to the left of z = 2 is approximately 0.9772.
Therefore, approximately 97.72% of the observations are less than 46.
b. To determine the approximate number of observations that are less than 46, we can multiply the percentage found in part a by the total number of observations (300).
Approximate number of observations = Percentage of observations * Total number of observations
Approximate number of observations = 0.9772 * 300
Approximate number of observations = 293.16
Approximately 293 observations are less than 46. (Rounded to the nearest whole number: 293)
To answer these questions, we can use the empirical rule, also known as the 68-95-99.7 rule. According to this rule, for a bell-shaped distribution:
1. Approximately 68% of the observations fall within one standard deviation of the mean.
2. Approximately 95% of the observations fall within two standard deviations of the mean.
3. Approximately 99.7% of the observations fall within three standard deviations of the mean.
a. To find the percentage of observations that are less than 46, we need to calculate the z-score for 46. The z-score represents the number of standard deviations an observation is from the mean.
The formula to calculate the z-score for an observation x, given the mean (μ) and standard deviation (σ), is:
z = (x - μ) / σ
Here, x = 46, μ = 40, and σ = 3. Plugging these values into the formula, we get:
z = (46 - 40) / 3 = 2
Using the empirical rule, we know that approximately 95% of the observations fall within two standard deviations of the mean. Since 46 is 2 standard deviations away from the mean, we can estimate that approximately 95% of the observations are less than 46.
b. To estimate the number of observations that are less than 46, we can use the percentage calculated in part (a) and multiply it by the total number of observations in the data set.
Approximately, 95% of the 300 observations are less than 46. To calculate the approximate number of observations, we multiply 0.95 by 300:
Number of observations = 0.95 * 300 ≈ 285
Therefore, approximately 285 observations are less than 46.