Find the area between x = 9.5 and x = 23 under a normal distribution curve with μ = 20 and σ = 4 .
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To find the area between x = 9.5 and x = 23 under a normal distribution curve with μ = 20 and σ = 4, you will need to use the standard normal distribution table or a statistical calculator.
Here's how you can use a standard normal distribution table:
1. Convert the given values to z-scores.
To find the z-score for x = 9.5, use the formula: z = (x - μ) / σ
z₁ = (9.5 - 20) / 4 = -2.375
To find the z-score for x = 23, use the same formula:
z₂ = (23 - 20) / 4 = 0.75
2. Look up the probabilities associated with the z-scores.
Using the standard normal distribution table, find the area to the left of z = -2.375 and z = 0.75.
The area to the left of z₁ is approximately 0.0089.
The area to the left of z₂ is approximately 0.7734.
3. Calculate the area between the two z-scores.
To find the area between z₁ and z₂, subtract the smaller area from the larger area:
Area = 0.7734 - 0.0089 = 0.7645
4. Interpret the result.
The area between x = 9.5 and x = 23 under the given normal distribution curve is approximately 0.7645, or 76.45%.
Note: You can also use a statistical calculator or software to calculate the area directly by inputting the required values, such as the mean, standard deviation, and the range of x-values.