SAT Math scores have a bell-shaped distribution with a mean...

SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of $114 .$ Source: College Board, 2010 (a) What percentage of SAT scores is between 401 and $629 ?$ (b) What percentage of SAT scores is less than 401 or greater than $629 ?$ (c) What percentage of SAT scores is greater than $743 ?$

SAT Math scores have a bell-shaped distribution with a mean of 515 and a standard deviation of $114 .$ Source: College Board, 2010
(a) What percentage of SAT scores is between 401 and $629 ?$
(b) What percentage of SAT scores is less than 401 or greater than $629 ?$
(c) What percentage of SAT scores is greater than $743 ?$

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Key Concepts

Empirical Rule

The Empirical Rule, also known as the 68-95-99.7 rule, is a guideline that indicates approximately 68% of data fall within one standard deviation of the mean, 95% within two, and 99.7% within three in a bell-shaped, normal distribution. This rule helps in estimating the percentage of data within specific intervals.

Standard Deviation

The standard deviation is a measure of the spread or dispersion of a set of values in a distribution. In a normal distribution, it quantifies how much the values deviate from the mean on average.

Z-Score

A Z-score is a standardized value that represents the number of standard deviations a data point is from the mean. It is used to calculate the probability of a score occurring within a normal distribution.

Normal Distribution

A normal distribution is a continuous probability distribution that is symmetric about its mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. It is characterized by its bell-shaped curve and defined by its mean and standard deviation.

Mean

The mean is the central value around which the data are distributed in a normal distribution. It represents the average of all data points and is used as the anchor point of the distribution.

Transcript

00:01 In this problem x will be sat math score, and we know that variable x follows the normal distribution.

00:13 Population mean mu is 515, and population standard deviation sigma is 114.

00:25 Task a.

00:29 We should find the percentage of sat scores that are between 401 and 629.

00:40 Now let's find the probability that x will be between 401 and 629.

00:53 This equals to probability that x is less than 629 minus probability that x is less than 401.

01:08 Next we obtain probability that sat score is less than 629 minus population mean 515 divided by standard deviation 114 minus probability that sat score is less than 401 minus 515 divided by 114.

01:41 And after calculations we obtain probability that sat score is less than 1 minus probability that sat score is less than minus 1.

01:55 And by using sat score table we can find the corresponding value of the probability.

02:05 So, we obtain 0 .8413 minus 0 .1587.

02:17 And this will be 0 .6826.

02:23 And now let's convert to percentage.

02:27 68 .26%.

02:30 So, this is the answer for the part a.

02:35 Next part b.

02:39 We should find the percentage of sat scores that are less than 401 or greater than 629.

02:52 This can be written as probability that x will be less than 401 and greater than 629.

03:05 This equals to 1 minus probability that x will be between 401 and 629...

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