To solve this question, we can use the properties of the standard normal distribution. For a normally distributed variable with mean (μ) and standard deviation (σ), we can find the area under the curve (the probability) to the left or right of a specific value by converting that value into a Z-score and then looking up the corresponding area in the standard normal distribution table (Z-table) or using a calculator with Z-score functionalities.
The Z-score is calculated using the following formula:
\[ Z = \frac{(X - \mu)}{\sigma} \]
where:
- \( X \) is the value for which we're finding the probability,
- \( \mu \) is the mean of the distribution,
- \( \sigma \) is the standard deviation of the distribution.
Let's calculate these areas step by step:
a) Find the area to the left of 26:
To find the Z-score for 26:
\[ Z_{26} = \frac{(26 - 21)}{7} = \frac{5}{7} \approx 0.7143 \]
Now, we look up the Z-score of 0.7143 in the Z-table or use a graphing calculator to find the area to the left:
The area to the left of Z = 0.7143 is approximately 0.762.
b) Find the area to the left of 16:
For the Z-score for 16:
\[ Z_{16} = \frac{(16 - 21)}{7} = \frac{-5}{7} \approx -0.7143 \]
The area to the left of Z = -0.7143 is approximately 0.238.
c) Find the area to the right of 17:
For the Z-score for 17:
\[ Z_{17} = \frac{(17 - 21)}{7} = \frac{-4}{7} \approx -0.5714 \]
The area to the left of Z = -0.5714 is approximately 0.284. To find the area to the right, we subtract from 1:
Area to the right of Z = -0.5714 is approximately 1 - 0.284 = 0.716.
d) Find the area to the right of 28:
For the Z-score for 28:
\[ Z_{28} = \frac{(28 - 21)}{7} = \frac{7}{7} = 1 \]
The area to the left of Z = 1 is approximately 0.8413. To find the area to the right, we subtract from 1:
Area to the right of Z = 1 is approximately 1 - 0.8413 = 0.1587.
e) Find the area between 16 and 30:
We will find the Z-scores for both 16 and 30 and then find the area to the left of each before subtracting one from the other.
For 16:
\[ Z_{16} = \frac{(16 - 21)}{7} = -0.7143 \]
(Already calculated in part b)
For 30:
\[ Z_{30} = \frac{(30 - 21)}{7} = \frac{9}{7} \approx 1.2857 \]
The area to the left of Z = 1.2857 is approximately 0.9007.
Now we subtract the area to the left of 16 from the area to the left of 30:
\[ Area_{16-30} = 0.9007 - 0.238 \]
\[ Area_{16-30} \approx 0.6627 \]
There you have all the areas you were looking for, each rounded to the nearest thousandth as needed:
a) ~0.762
b) ~0.238
c) ~0.716
d) ~0.1587
e) ~0.6627
Please note that the exact values can differ slightly depending on the precision of the Z-table or graphing calculator used.
