To find the proportion of phones that have between 45 and 83 apps, we will use the properties of the normal distribution. Given that the number of apps is normally distributed with a mean (\(\mu\)) of 80 and a standard deviation (\(\sigma\)) of 20, we need to calculate the z-scores for both values (45 and 83).
The formula to calculate the z-score is:
\[ z = \frac{(X - \mu)}{\sigma} \]
Step 1: Calculate the z-score for 45 apps
For \(X = 45\):
\[ z_{45} = \frac{(45 - 80)}{20} = \frac{-35}{20} = -1.75 \]
Step 2: Calculate the z-score for 83 apps
For \(X = 83\):
\[ z_{83} = \frac{(83 - 80)}{20} = \frac{3}{20} = 0.15 \]
Step 3: Look up the z-scores in the standard normal distribution table
Using the standard normal distribution table or a calculator:
- For \(z = -1.75\), the area to the left is approximately \(0.0401\).
- For \(z = 0.15\), the area to the left is approximately \(0.5596\).
Step 4: Calculate the proportion of phones with apps between 45 and 83
To find the proportion of phones with between 45 and 83 apps, subtract the area to the left of \(z = -1.75\) from the area to the left of \(z = 0.15\):
\[ P(45 < X < 83) = P(Z < 0.15) - P(Z < -1.75) \]
\[ P(45 < X < 83) = 0.5596 - 0.0401 = 0.5195 \]
Final Result
The proportion of phones that have between 45 and 83 apps is approximately \(0.5195\).
Thus, rounding to four decimal places:
\[ \boxed{0.5195} \]
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