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A set of philosophy exam scores are normally distributed with a mean of 40 points points and a standard deviation of three. Points ludwig got a score of 475 points the exam. What proportion of exam scores are higher than ludwigs score give your answer correct to four decimal places.
So lets just visualize. Whats going on here. So the scores are normally distributed.
So it would look like this so the distribution would look something like that trying to make that pretty symmetric looking the mean is 40 points. So that would be 40 points right over there standard deviation is three points. So this could be one standard deviation above the mean that would be one standard deviation below the mean and once again this is just very rough and so this would be 43.
This would be 37 right over. Here and they say ludwig got a score of 475 points. On the exam.
So ludwigs score is going to be someplace around. Here so ludwig got a 475. On the exam and theyre saying.
What proportion of exam scores are higher than ludwigs score. So what we need to do is figure out what is the area under the normal distribution curve that is above 475. So the way we will tackle this is we will figure out the z score for 475.
How many standard deviations above. The mean is that then we will look at a z table to figure out what proportion is below that because thats what z tables give us they give us the proportion that is below a certain z score. And then we can take one minus that to figure out the proportion that is above remember the entire area under the curve is one so if we can figure out this orange area and take one minus.
That were gonna get the red area. So lets do that so first of all lets figure out the z score for. 475.
So lets see we would take 475. And we would subtract the mean so this is his score. Well subtract the mean minus.
40 we know what thats gonna be thats 75. Thats how much more above the mean. But how many standard deviations is that well each standard deviation is.
Three so whats 75. Divided by three this just means the previous. Answer divided by.
Three so he is 25. Standard deviations above the. Mean so the z score here.
Z. Score here is a positive 25. If he was below the mean it would be a negative.
So now we can look at a z table to figure out what proportion is less than 25. Standard deviations above the mean. So thatll give us that orange.
And then well subtract that from one. So lets get our z table so here we go and weve already done this in previous videos. But whats going on here is this left column gives us our z score up to the tenths place.
And then these other columns give us the hundredths. Place so what we want to do is find. 25 right over here on the.
Left and its actually gonna be. 250 theres zero hundredths. Here so we want to look up 250 let me scroll my z table so im gonna go down to 25.
Alright i think i am there so what i have here so i have 25. So i am going to be in this row. And its now scrolled off.
But this first column we saw this is the hundredths place and this zero hundredths and so 250. Puts us right over. Here.
Now you might be tempted to say okay. Thats the proportion that scores higher than ludwig. But youd be wrong.
This is the proportion that scores lower than ludwig. So what we wanna do is take one minus this value. So let me get my calculator out again.
So what im going to do is im going to take one. Minus this one minus. 09938 is equal to now.
This is so this is the proportion that scores less than ludwig one minus. That is gonna be the proportion that scores more than him. The reason.
Why we have to do this is because the z table gives us the proportion less than a certain z. Score so this gives us right over. Here.
00062 so thats the proportion if you thought of it in. Percent it would be 062 scores higher than ludwig now that makes sensecause ludwig scored over two standard deviations two and a half standard deviations above the. Mean so our answer.
Is. 00062 so this is going to be. 00062 thats the proportion of exam scores higher than ludwigs score.
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