To conduct a goodness-of-fit test to determine whether the sample appears to have been selected from a normal distribution, we can use the mean and the standard deviation provided to determine the boundaries of the intervals under the normal distribution.
Given:
- Sample size \( n = 40 \)
- Sample mean \( \mu = 63 \)
- Sample standard deviation \( \sigma = 12 \)
Since we want to find the fourth interval, we need to determine the z-scores associated with the normal distribution.
For a standard normal distribution:
- The first interval (mean - 3σ to mean - 2σ) is approximately the range containing the lowest 2.5% of the data.
- The second interval (mean - 2σ to mean - 1σ) is between 2.5% to 15.87%.
- The third interval (mean - 1σ to mean + 1σ) represents the middle 68.27% of the data.
- The fourth interval (mean + 1σ to mean + 2σ) generally contains the data from around 84.13% up to 97.5%.
Calculating the boundaries:
- Mean - 3σ = \( 63 - 3(12) = 27 \)
- Mean - 2σ = \( 63 - 2(12) = 39 \)
- Mean - 1σ = \( 63 - 12 = 51 \)
- Mean + 1σ = \( 63 + 12 = 75 \)
- Mean + 2σ = \( 63 + 2(12) = 87 \)
Thus, the intervals with their corresponding values would be:
- First interval: (27, 39)
- Second interval: (39, 51)
- Third interval: (51, 63)
- Fourth interval: (63, 75)
To identify the fourth interval:
- The fourth interval contains values from \( \mu \) (mean = 63) to \( \mu + 2σ \) (75).
The possible intervals presented are:
- 54.96 – 59.16
- 49.2 – 54.96
- 58.26 – 62.54
- 59.16 - 63
- 64.26 – 66.28
Based on our calculations, the fourth interval corresponding to values from mean + 1σ to mean + 2σ is approximately: \[ 63 \text{ to } 75 \]
Since the closest boundary from the choices provided that fits in between the generated intervals would be 64.26 – 66.28, because it represents the interval immediately following the mean of 63.
Thus, the final answer is:
64.26 – 66.28