00:01
For this problem, we're told that the reading speed of second graders is approximately normal.
00:06
The mean mu is given by 84 words per minute, and the standard deviation sigma is given by 11 words per minute.
00:16
We're going to let x be our random variable for the population here, and for part a, we want the probability that a randomly selected student read more than 97 words per minute.
00:29
So that's the probability that x is greater than 97.
00:35
So since x is approximately normally distributed, let me draw a normal curve down here.
00:43
So the mean is here in the middle, that's 84, and we want all the area to the right of 97.
00:49
So that will look something like this under the curve, the area i'm drawing in red going all the way up to plus infinity.
00:56
That area i've drawn is equal to this probability.
00:59
Now we're told that x is approximately normally distributed, so we can work this out with the cumulative function for normal distributions.
01:10
You can use any tool you like for this, tables, programs, calculators.
01:14
I'm going to use a ti -84, which has the normal cdf function.
01:21
This function takes in four inputs and gives you your probability as an output.
01:26
So the first thing you need is the lower limit of your probability, where the area under the curve starts.
01:32
It's 97.
01:34
Then you need the upper limit under the curve of your probability, where the area under the curve ends.
01:39
We want everything to the right of 97, so that's the upper limit of plus infinity, but we don't have a button for that on the calculator, so we just have to use a big positive number like 10 to the power of 99, which is written as e to the power of 99 on the calculator.
01:55
Next you need the mean and standard deviation of the distribution.
01:58
So that's 84 for the mean and 11 for the standard deviation.
02:02
Let me just go put this into my calculator and we'll see what this probability is equal to.
02:13
So it looks like this is 0 .118624 decimal places.
02:21
So there is our answer for part a.
02:25
Next, for part b, we're going to take a random sample of 17 students.
02:31
So let's say n is 17, and we want to calculate a probability for the sampling distribution.
02:40
So we need to work out a few more parameters.
02:43
First, we want the mean for our sampling distribution.
02:46
On that, we use some x -bar.
02:48
We use x -bar for our random variable.
02:51
That's very generally just equal to the population mean, which is 84.
02:56
And the standard error, or standard deviation, for the sampling distribution.
03:01
That has a general formula as well.
03:05
Sigma over root n, the population standard deviation divided by the square root of your sample size.
03:12
So for us, that's 11 over root 17.
03:17
And since our original population is approximately normally distributed, we can also treat x by approximately normally distributed.
03:29
So we are back to working with the normal distribution for these probabilities.
03:37
So for part b, we want the probability that our sample mean will be more than 92 words per minute.
03:46
So that's a probability that x -bar is greater than 92.
03:49
In terms of area under the curve, we can use roughly the same picture.
03:54
We just have to imagine that the curve has become a bit more narrow due to the smaller standard deviation we'll have.
04:04
And the area of the curve will look something like this...