Suppose the age that children learn to walk is normally distributed with mean 11 months and standard deviation 2.5 month. 34 randomly selected people were asked what age they learned to walk. Round all answers to 4 decimal places where possible.
a. What is the distribution of
b. What is the distribution of $\bar{x} ? \bar{x}-N$
c. What is the probability that one randomly selected person learned to walk when the person was between 10.6 and 11.4 months old?
d. For the 34 people, find the probability that the average age that they learned to walk is between 10.6 and 11.4 months old.
e. For part d), is the assumption that the distribution is normal necessary? Yes - No $0^{\circ}$
f. Find the $I Q R$ for the average first time walking age for groups of 34 people.
$Q 1=\square$ months
$Q 3=\square$ months
IQR: $\square$ months

Step 1 ：The problem states that the age that children learn to walk is normally distributed with a mean of 11 months and a standard deviation of 2.5 months. This can be represented as \(N(11, 2.5)\).

Step 2 ：The distribution of the sample mean, denoted as \(\bar{x}\), is also normally distributed. The mean is the same as the population mean, but the standard deviation is the population standard deviation divided by the square root of the sample size. Therefore, the distribution of \(\bar{x}\) is \(N\left(11, \frac{2.5}{\sqrt{34}}\right)\).

Step 3 ：The probability that a randomly selected person learned to walk when the person was between 10.6 and 11.4 months old can be calculated using the cumulative distribution function (CDF) of the normal distribution. The result is approximately \(0.1271\).

Step 4 ：The probability that the average age that 34 people learned to walk is between 10.6 and 11.4 months old can also be calculated using the CDF, but with the distribution of the sample mean. The result is approximately \(0.6492\).

Step 5 ：For part d), the assumption that the distribution is normal is necessary because the sample size is not large enough for the Central Limit Theorem to guarantee that the distribution of the sample mean is approximately normal.

Step 6 ：The interquartile range (IQR) for the average first time walking age for groups of 34 people is the range between the first quartile (Q1) and the third quartile (Q3). These can be calculated using the inverse of the CDF (also known as the quantile function) of the distribution of the sample mean. The result is approximately \(0.5784\) months, with \(Q1\) approximately \(10.7108\) months and \(Q3\) approximately \(11.2892\) months.