(a) s² = å (x – x )² / (n-1).

(b) s² = [ å x² - ( å x )²/ n ] / (n-1)

deviation

Cookie x =# chips (x – x ) (x – x )² x²

1. 5 5 – 7 = -2 4 25

2 11 11 – 7 = 4 16 121

3 6 6 – 7 = -1 1 36

4 8 8 – 7 = 1 1 64

5 5 5 – 7 = -2 4 25

å x = 35 å (x – x )² = 26 å x² = 271

Using (a), s² = å (x – x )² / (n-1) = 26 / (5-1) = 26 / 4 = 6.5

Using (b), s² = [ å x² - ( å x )²/ n ] / (n-1) = [271 – (35)² / 5] / (5-1) = (271 – 245 ) / 4 = 26/4 = 6.5

The sample standard deviation s is the square root of the sample variance:

s = / s² = / 6.5 = 2.55

The quantities s and s² are statistics, that is, they are numerical values calculated from a sample.

1. Find the sample mean x , the sample variance s² , and the sample standard deviation (sd) of the following data on the number x of dates 4 males had in the past month, using both formulas:

 

Male 1 2 3 4

 

x = # dates 3 2 6 1

 

Calculate s and s² in Minitab to check your calculations.

 

2. Find the sample mean, sample variance, and sample standard deviation (sd) for the following number x of tattoos that 3 students randomly selected had:

 

Student George Roberta Mary

 

x = # Tattoos 2 4 5

 

Also, find the median. Mean = ______; Median = ______; s² = ______; s = ______ .

 

Hint: it is easier to use the second formula above to find s². Calculate s and s² in Minitab to check your calculations.

 

 

 

 

 

 

 

 

 

3. Change the last number—5—in ii above, to 20 and recalculate the same 4 quantities. Then discuss briefly the effect of putting in a value far from the middle value (median). The median is a statistic that is resistant to outliers, in the sense that it’s values are not greatly affected by outliers. The Interquartile Range is also a resistant statistic, whereas the mean and sample standard deviation/variance are not resistant. Calculate s and s² in Minitab to check your calculations.

1. The weights of students in the Physical Measurements Data Set have the following statistics:

 

Males: Average = 175.92, SD s= 27.40

Females: Average = 130.46, SD s = 17.62

 

Using the Empirical Rule, find the endpoints of the interval of weights which should contain about 68%, 95%, and 99.7% of the weights for both males and females. It turns out that the actual percents in the three intervals for males are 72.24%, 97.34%, and 98.86%, while for females the actual percents are 75.2%, 95.73%, and 98.58%. See section 2.6 in Heckard/Utts for a similar example on heights.

 

2. The probability distribution of the number X of times students change their major(s) is as follows:

x p(x)

 

0. .50
1. .30
2. .10
3. .07
4. .03

i. What proportion of students change their majors exactly 2 times? Answer: ____

2. What proportion of students change their majors at most 2 times (at most 2 means 2 or fewer times)?

Answer: ____

 

iii. Find the mean m of the probability distribution.(show work above) Answer: ____

iv, Find the population variance and SD s ² and s , respectively. Answer: ____ and ____

4. The probability distribution of the number X of days/week college students have at least one alcoholic drink is given below. Find the mean m , and variance s ² and SD s of X.

 

x p(x)

 

1. .15
2. .15
3. .35
4. .25
5. .08
6. .02

5. Write a short paragraph explaining the difference between the sample mean and the population mean, using the random variable X in problem 4 above. Assume that the population consists of the numbers 0, 1, 2, 3, 4, and 5 corresponding to the number of days per week each of the 30000 PSU students have at least one alcoholic beverage. Assume also that one has a sample of size 800 from this population.

 

 

 

 

 

 

 

6. Write a short paragraph explaining the difference between the sample sd and the population sd, using the random variable X in problem 4 above. Assume that the population consists of the numbers 0, 1, 2, 3, 4, and 5 corresponding to the number of days per week each of the 30000 PSU students have at least one alcoholic beverage. Assume also that one has a sample of size 800 from this population. Also answer the following questions::

1. Is the population mean a fixed quantity? Or does it change with each sample that is taken?
2. Is the sample mean a fixed quantity? Or does it change with each sample that is taken?
3. What is the formula used to calculate the sample sd?
4. What is the formula used to calculate the population sd?
5. Is the sample mean a statistic or a parameter?
6. Is the population mean a statistic or parameter?
7. What is the relation between the sample mean and population mean?
8. What is the relation between the sample sd and the population sd?

Short paragraph:

Answer to questions:

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2. ------------------------------------------

 

3. Formula:

 

 

4. Formula:

 

 

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