If scores are normally distributed with a mean of 3.06 and a standard deviation of 0.84, what percent of the scores is: (a) greater than 3.9? (b) less than 2.22? (c) between 2.22 and 3.9?
To find the percent of scores that fall into each category, we can use the standard normal distribution table or a calculator that has built-in functions for computing these probabilities. I will explain how to calculate each of the probabilities using the standard normal distribution table.
(a) To find the percent of scores that are greater than 3.9, we need to calculate the area under the curve to the right of 3.9.
Step 1: Standardize the value using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
z = (3.9 - 3.06) / 0.84
z = 0.94
Step 2: Look up the area to the right of 0.94 in the standard normal distribution table (also known as the Z-score table). The value you find is the percentage of scores greater than 3.9.
According to the standard normal distribution table, the area to the right of 0.94 is approximately 0.1736. Therefore, approximately 17.36% of the scores are greater than 3.9.
(b) To find the percent of scores that are less than 2.22, we need to calculate the area under the curve to the left of 2.22.
Step 1: Standardize the value using the formula z = (x - μ) / σ.
z = (2.22 - 3.06) / 0.84
z = -1.00
Step 2: Look up the area to the left of -1.00 in the standard normal distribution table. The value you find is the percentage of scores less than 2.22.
According to the standard normal distribution table, the area to the left of -1.00 is approximately 0.1587. Therefore, approximately 15.87% of the scores are less than 2.22.
(c) To find the percent of scores that are between 2.22 and 3.9, we need to calculate the area under the curve between these two values.
Step 1: Standardize both values using the formula z = (x - μ) / σ.
For 2.22:
z1 = (2.22 - 3.06) / 0.84
z1 = -1.00
For 3.9:
z2 = (3.9 - 3.06) / 0.84
z2 = 0.94
Step 2: Find the area to the left of each value using the standard normal distribution table.
For z1 = -1.00, the area to the left is approximately 0.1587.
For z2 = 0.94, the area to the left is approximately 0.8279.
Step 3: Calculate the percentage of scores between the two values by subtracting the smaller area from the larger area.
Percentage = Larger area - Smaller area
Percentage = 0.8279 - 0.1587
Percentage ≈ 0.6692
Therefore, approximately 66.92% of the scores are between 2.22 and 3.9.