Test 1 - Math 1342

Test 1 - Math 1342

1.

Summary statistics:

Statistic

Std. Error

Grams of protein in fast food sandwiches

Mean

27.6250

1.52151

95% Confidence Interval for Mean

Lower Bound

24.5475

Upper Bound

30.7025

5% Trimmed Mean

27.2222

Median

26.5000

Variance

92.599

Std. Deviation

9.62286

Minimum

12.00

Maximum

57.00

Range

45.00

Interquartile Range

13.50

Skewness

.723

.374

Kurtosis

.924

.733

2.

Grams of protein in fast food sandwiches Stem-and-Leaf Plot

 Frequency    Stem &  Leaf

     3.00        1 .  224

     4.00        1 .  5589

     9.00        2 .  001223344

    10.00        2 .  5666777799

     4.00        3 .  0134

     5.00        3 .  55558

     4.00        4 .  0234

     1.00 Extremes    (>=57)

 Stem width:     10.00

 Each leaf:       1 case(s)

3.

The data is comprised on even numbers items; thus, the middle two numbers were averaged. The two numbers that were averaged are 26 and 27.

4.

From the box plot, it is observed that the median is shifted toward Q1.

5.

Min

Max

Q1

Q3

IQR

12

57

21.25

34.75

13.5

Min Max Q1 Q3 IQR

21 52 30 46 16

1.5 * IQR = 1.5 * 13.5 = 20.25

Q1 - (1.5 * IQR) = 21.25 - 20.25 = 1

Q3 + (1.5 * IQR) = 34.75 + 20.25 = 55

Thus, if there are any outliers then they will be outside the range of 1 to 55.

Min = 12 which is not lower than 1 and

Max = 57 which is higher than 55 thus, the data has 2 outliers.

6.

Class Limits

Class Boundaries

Frequency

Midpoints

Freq * Midpoints

Cumulative Freq.

6 - 13

5.5 - 13.5

2

9.5

19

5

14 - 21

13.5 - 21.5

4

17.5

70

15

22 - 29

21.5 - 29.5

10

25.5

255

40

30 - 37

29 .5- 37.5

12

33.5

402

70

38 - 45

3.5 - 45.5

6

41.5

249

85

46 - 53

45.5 - 53.5

5

49.5

247.5

97.5

54 - 61

53.5 - 61.5

1

57.5

57.5

100

7.

Grouped Mean

= Sum of Column of Freq * Midpoints / Sample Size

= 1300 / 40

= 32.5

The mean from the original ungrouped data is 27.6 and the grouped mean is 32.5. The numbers are not the same and the reason that they are not is that the grouped table has lost some of the precision due to the midpoints being used to represent the data.

8.

9.

10.

Summary statistics:

Statistic

Std. Error

data

Mean

9.5556

1.19701

95% Confidence Interval for Mean

Lower Bound

7.0301

Upper Bound

12.0810

5% Trimmed Mean

9.4506

Median

8.5000

Variance

25.791

Std. Deviation

5.07847

Minimum

1.00

Maximum

20.00

Range

19.00

Interquartile Range

5.50

Skewness

.710

.536

Kurtosis

.516

1.038

a. z score for 13 = (value - mean)/std = (13 - 9.55)/ 5.078 = 0.6794

b. z score for 5 = (5 - 9.55)/ 5.078 = -0.896

For next 2 problems we need the data sorted

Data

1

3

5

6

7

7

8

8

8

9

9

10

11

12

13

15

20

20

Position

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

c. Percentile Rank

How many numbers are under 12?

13 of them

Percentile = (13 + 0.5) / 15 * 100 = 13.5/15 * 100 = 90th

d. 40th Percentile

c = (15 * 40) / 100 = 6

6 is NOT the answer, 6 is the position where the 40th percentile is at. Position 6 number is 7.

Data

1

3

5

6

7

7

8

8

8

9

9

10

11

12

13

15

20

20

Position

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

11.

From the theory we know the interval is always

(mean - k * standard deviation, mean + k * standard deviation )

So matching

mean = 25, standard deviation = 4, k = unknown

(25 - 4k, 25 + 4k) = (19, 31)

So

25 - 4k = 19 and solve for k

- 4k = - 6

k = 6/4 = 2

The proportion is

1 - 1/k ^ 2 = 1 ...