A random sample of size n = 40 is taken. Conduct a goodness of fit test to see whether this sample appears to have been selected from a normal distribution. The mean and the standard deviation of the sample are respectively 63 and 12. The boundary of the first interval is:

To conduct a goodness-of-fit test to check if a sample comes from a normal distribution, we can use a chi-squared test. However, first, we need to define the intervals based on the normal distribution parameters.

Given:
- Sample size \( n = 40 \)
- Sample mean \( \mu = 63 \)
- Sample standard deviation \( \sigma = 12 \)

For a normal distribution, we typically create intervals (bins) around the mean using standard deviations. A common approach is to set the intervals based on:

1. \( \mu - 3\sigma \) to \( \mu + 3\sigma \) covers about 99.7% of the data in a normal distribution.
2. We can create several intervals to perform the goodness-of-fit test (usually we create at least 5 intervals).

Calculating the boundaries for the first interval:

1. Let's find the boundaries if we are creating intervals centered around the mean.
2. The first interval might span from \( \mu - 1\sigma \) to \( \mu + 1\sigma \):
- Lower boundary: \( \mu - \sigma = 63 - 12 = 51 \)
- Upper boundary: \( \mu + \sigma = 63 + 12 = 75 \)

Thus, the first interval could be \( (51, 75) \).

The lower boundary of the first interval is \( 51 \).

If further details are required, such as confidence intervals or specific interval divisions for the chi-squared test, please provide that, and I'll be glad to elaborate!