The playing time X of classical CDs has the normal distribution with mean 54 and standard deviation 5; the N(54, 5) distribution. What is the relative frequency of classical CDs with playing time X between 49 and 69 minutes?
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To find the relative frequency of classical CDs with playing time between 49 and 69 minutes, we need to calculate the area under the normal distribution curve between these two values.
Step 1: Standardize the values
In order to use the normal distribution tables, we need to convert the values from minutes to a standard score (also known as z-scores) using the formula:
z = (x - μ) / σ
Where:
x = value from the distribution (in this case, 49 and 69)
μ = mean of the distribution (54)
σ = standard deviation of the distribution (5)
For x = 49:
z1 = (49 - 54) / 5 = -1
For x = 69:
z2 = (69 - 54) / 5 = 3
Step 2: Look up the standard scores
Use a standard normal distribution table or a statistical calculator to find the areas under the curve corresponding to the z-scores -1 and 3.
Looking up the standard normal distribution table, we find:
For z = -1, the area to the left is 0.1587.
For z = 3, the area to the left is 0.9987.
Step 3: Calculate the relative frequency
To find the relative frequency, subtract the area to the left of the lower z-score from the area to the left of the higher z-score:
Relative frequency = P(49 ≤ X ≤ 69) = P(X ≤ 69) - P(X ≤ 49) = 0.9987 - 0.1587 = 0.84
Therefore, the relative frequency of classical CDs with playing time between 49 and 69 minutes is 0.84, or 84%.
To find the relative frequency of classical CDs with playing time X between 49 and 69 minutes, we need to calculate the area under the normal distribution curve between these two values.
1. Calculate the z-scores for the lower and upper limits:
- For 49 minutes: z1 = (49 - 54) / 5 = -1
- For 69 minutes: z2 = (69 - 54) / 5 = 3
2. Use a standard normal table or a calculator to find the cumulative probabilities:
- For z1 = -1, the cumulative probability is P(Z < -1) = 0.1587
- For z2 = 3, the cumulative probability is P(Z < 3) = 0.9987
3. Calculate the relative frequency:
- Relative frequency = P(49 < X < 69) = P(-1 < Z < 3)
- This can be calculated by subtracting the cumulative probabilities:
P(-1 < Z < 3) = P(Z < 3) - P(Z < -1) = 0.9987 - 0.1587 = 0.84
Therefore, the relative frequency of classical CDs with playing time X between 49 and 69 minutes is 0.84 or 84%.