Exercise 3-59 Algo
Observations are drawn from a bell-shaped distribution with a mean of 210 and a standard deviation of 5. There are 1,900 observations in the data set.
a. Approximately what percentage of the observations are less than 220? (Round your answer to 1 decimal place.)
b. Approximately how many observations are less than 220? (Round your answer to the nearest whole number.)
a. To find the percentage of observations that are less than 220, we need to find the area under the bell-shaped distribution curve to the left of 220.
To do this, we can use a standard normal distribution table or a calculator.
Using a standard normal distribution table, we can find the z-score corresponding to a value of 220:
z = (x - mean) / standard deviation
z = (220 - 210) / 5
z = 2
Looking up the z-score of 2 in the standard normal distribution table, we find that the area to the left of 2 is approximately 0.9772.
So, approximately 97.7% of the observations are less than 220.
b. To find the approximate number of observations that are less than 220, we can multiply the percentage from part a by the total number of observations.
Approximately, 97.7% * 1,900 = 1,853 observations are less than 220.
a. To find the percentage of observations that are less than 220, we can use the Z-score formula:
Z = (X - mean) / standard deviation
where X is the value we want to find the percentage for.
First, let's calculate the Z-score for 220 using the given mean of 210 and standard deviation of 5:
Z = (220 - 210) / 5
Z = 10 / 5
Z = 2
Next, we can use the Z-score to find the corresponding percentage using a Z-table or a calculator.
Looking up the Z-score of 2 in a Z-table, we find that the area to the left of 2 is approximately 0.9772.
Since the area to the left of a specific Z-score represents the percentage of observations that are less than that value, the percentage of observations that are less than 220 is approximately 97.7%.
b. To find the number of observations that are less than 220, we can use the formula:
Number of observations = Percentage x Total number of observations
Percentage = 0.9772
Total number of observations = 1,900
Number of observations = 0.9772 * 1,900
Number of observations ≈ 1859.88
Rounding to the nearest whole number, we can say that approximately 1860 observations are less than 220.