00:01
It's given here that the ages of students at a certain elementary school are uniformly distributed between 6 and 10 years old.
00:10
So here age is the random variable x.
00:16
And so for part a we're asked for the distribution of x, so it's uniformly distributed on the interval going from 6 to 10.
00:26
And for part b we're asked if 47 children from the school are surveyed, what is the sampling distribution of sample means? so here we are drawing a sample of size 47 from the population of students.
00:43
And since 47 is large, for example, it's bigger than 30, this means that the central limit theorem comes into play, and this means that sample means are approximately normally distributed.
01:05
And the mean of sample means is equal to the mean of the population from which the samples are drawn.
01:14
For a uniform random variable, the mean is equal to a plus b over 2, where a is the lower end of the range and b is the upper end of the range.
01:31
And so this is 8.
01:37
And the standard deviation sample means is equal to the standard deviation of the population over the square root of the sample size.
01:51
Now the standard deviation for a discrete uniform random variable is the square root of the following.
02:14
So here we have the square root of 10 minus 6 plus 1 squared minus 1 over 12, which comes out to 1 .4142 approximately.
02:40
So this gives us a standard deviation for sample means of 0 .2063.
02:49
And so we can say that sampling distribution of sample means is normal with the mean of 8 and the standard deviation of 0 .206...