Question

# 1. Calculate the mean and standard deviation for the numbers 1 – 20. ​i.e.) 1, 2,...

1. Calculate the mean and standard deviation for the numbers 1 – 20.

​i.e.) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20

​Note: The set of numbers will be considered the population.

​µ = _______​​​σ = ________

2. With your calculator, randomly generate 5 numbers from the numbers 1 – 20, 30 times.

​Use:​[MATH]>>>PRB #5

​​RandInt(1,20,5)​[ENTER]​​Note: You cannothave the same number repeated in the group of 5.

​​​​​​For example: You cannot have {2,2,10,13,8} where the 2 is repeated.

After you generate your first 5 numbers, write them down in the space below, then hit [ENTER] again to generate

another 5 numbers. When you finish, you should have 30 groups of 5numbers ranging from 1 – 20.

3. Calculate the mean for each of the 30 groups and write them in the space below. You should have 30 sample means

when you finish.

4. Calculate the mean and standard deviation for the 30 sample means. Place your values in the space below and

label your values with the appropriate symbols. Note: Enter your 30 sample means into L1 in your calculator

and do 1-var stat L1 to get the mean and standard deviation.

5.   Using your values from part 1 and part 4, test the 3 properties of the Distribution of Sample Means:

​a.) Write the property for Mean of the Sample Means. Show the values for your means and state whether they

​   approximate the property.

​b.)   Write the property for the Standard Deviation of the Sample Means. Show the values for your standard

​   deviations and state whether they approximate the property.

​c.)   On a piece of graph paper, draw a graph (histogram) to approximate the population, then draw a 2nd

​   histogram to approximate the 30 sample means.  ( Hint: Create a frequency distribution for each set of

​   data – population and sample means – then draw a histogram for each frequency distribution.)

Mean ( ) = 10.5

standard deviation ( ​​​σ ) = 5.7663

Explanation :

( 1 )

• Mean ( ) = Sum of terms / Number of terms

=  210 / 20

= 10.5

• standard deviation ( ​​​σ ) :

Create the following table.

So mean = 10.5

 data data-mean (data - mean)2 1 -9.5 90.25 2 -8.5 72.25 3 -7.5 56.25 4 -6.5 42.25 5 -5.5 30.25 6 -4.5 20.25 7 -3.5 12.25 8 -2.5 6.25 9 -1.5 2.25 10 -0.5 0.25 11 0.5 0.25 12 1.5 2.25 13 2.5 6.25 14 3.5 12.25 15 4.5 20.25 16 5.5 30.25 17 6.5 42.25 18 7.5 56.25 19 8.5 72.25 20 9.5 90.25

∑(data - mean)2= 665

σ = Sqrt ( ∑(data - mean)2 / n )

= Sqrt ( 665 / 20 )

= Sqrt ( 33 .25 )

= 5.76628130

~ 5.7663 ( approximately)

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