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.)
To find the percentage of observations that are less than 46, we need to calculate the z-score for 46 and use the standard normal table.
The z-score for 46 is calculated as: (46 - 40) / 3 = 2.
From the standard normal table, we can find that the percentage of observations that are less than 2 (z-score of 2) is approximately 97.7%.
a. Therefore, approximately 97.7% of the observations are less than 46.
To find the number of observations that are less than 46, we can multiply the percentage by the total number of observations.
b. Approximately 97.7% x 300 = 293 observations are less than 46. Rounded to the nearest whole number, approximately 293 observations are less than 46.
To find the percentage of observations that are less than 46, we will use the standard normal distribution table.
a. Calculate the z-score using the formula:
z = (x - mean) / standard deviation
Substituting the given values:
z = (46 - 40) / 3 = 2
Look up the z-score of 2 in the standard normal distribution table. The cumulative probability associated with a z-score of 2 is approximately 0.9772.
Therefore, approximately 97.7% of the observations are less than 46.
b. To find the number of observations less than 46, multiply the percentage found in part (a) by the total number of observations:
Number of observations = Percentage * Total number of observations
Number of observations = 0.9772 * 300 ≈ 293
Approximately 293 observations are less than 46.