Question

Create a vector named iq of 100 elements of N(100,20) (normal with a mean of 100 and std dev of 20) data. 6. add 10 to every element of iq 7. calculate the mean and std dev of iq?

Answer 1

Answer:

See code and explanation below.

Step-by-step explanation:

For this case we can use the following R code to create a vector of 100 elements from a normal distribution given by:

$X \sim N (\mu = 100. \sigma= 20)$

The function rnorm creates a sample data from the normla distribution with the mean and deviation provided

> normal<-rnorm(100,mean = 100,sd=20)

We can visualize the data like this

> normal

 [1]  87.47092 103.67287  83.28743 131.90562 106.59016  83.59063 109.74858 114.76649 111.51563

[10]  93.89223 130.23562 107.79686  87.57519  55.70600 122.49862  99.10133  99.67619 118.87672

[19] 116.42442 111.87803 118.37955 115.64273 101.49130  60.21297 112.39651  98.87743  96.88409

[28]  70.58495  90.43700 108.35883 127.17359  97.94425 107.75343  98.92390  72.45881  91.70011

[37]  92.11420  98.81373 122.00051 115.26351  96.70953  94.93277 113.93927 111.13326  86.22489

[46]  85.85010 107.29164 115.37066  97.75308 117.62215 107.96212  87.75947 106.82239  77.41274

[55] 128.66047 139.60800  92.65557  79.11731 111.39439  97.29891 148.03236  99.21520 113.79479

[64] 100.56004  85.13454 103.77585  63.90083 129.31110 103.06507 143.45223 109.51019  85.80107

[73] 112.21453  81.31805  74.92733 105.82892  91.13416 100.02211 101.48683  88.20958  88.62663

[82]  97.29643 123.56174  69.52866 111.87892 106.65901 121.26200  93.91632 107.40038 105.34198

[91]  89.14960 124.15736 123.20805 114.00427 131.73667 111.16973  74.46816  88.53469  75.50775

[100]  90.53199

Then we can add 10 to each element of the vector like this:

> normal1<-normal+10

And we can visualize the results like this

> normal1

 [1]  97.47092 113.67287  93.28743 141.90562 116.59016  93.59063 119.74858 124.76649 121.51563

[10] 103.89223 140.23562 117.79686  97.57519  65.70600 132.49862 109.10133 109.67619 128.87672

[19] 126.42442 121.87803 128.37955 125.64273 111.49130  70.21297 122.39651 108.87743 106.88409

[28]  80.58495 100.43700 118.35883 137.17359 107.94425 117.75343 108.92390  82.45881 101.70011

[37] 102.11420 108.81373 132.00051 125.26351 106.70953 104.93277 123.93927 121.13326  96.22489

[46]  95.85010 117.29164 125.37066 107.75308 127.62215 117.96212  97.75947 116.82239  87.41274

[55] 138.66047 149.60800 102.65557  89.11731 121.39439 107.29891 158.03236 109.21520 123.79479

[64] 110.56004  95.13454 113.77585  73.90083 139.31110 113.06507 153.45223 119.51019  95.80107

[73] 122.21453  91.31805  84.92733 115.82892 101.13416 110.02211 111.48683  98.20958  98.62663

[82] 107.29643 133.56174  79.52866 121.87892 116.65901 131.26200 103.91632 117.40038 115.34198

[91]  99.14960 134.15736 133.20805 124.00427 141.73667 121.16973  84.46816  98.53469  85.50775

[100] 100.53199

The sample mean is given by:

$\bar X = \frac{\sum_{i=1}^n X_i}{n}$

And the sample deviation by:

$s =\sqrt{\frac{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}}$

If we want to calculate the mean and standard deviation for the original data we can do this:

> mean(normal)

[1] 102.1777

And the standard deviation with:

> sd(normal)

[1] 17.96399

Other way to calculate the deviation is:

> sqrt(sum((normal-mean(normal))^2/(length(normal)-1)))

[1] 17.96399