i need help answering this question

Question

QUESTION 1 [20]
A manufacturer of children toys claims that less than 3% of his products are defective. When 500 toys were drawn from a large production run, 5% were found to be defective.
a) Define the
i. Population of interest [4]
ii. Sample [4]
iii. Parameter [4]
iv. Statistics [4]
b) Does the value 3% refer to the parameter or the statistics? Why? [2]
c) Does the value 5% refer to the parameter or the statistics? Why? [2]

QUESTION 2 [20]
The grades on a statistics exam for a sample of 40 students are as follows:
63 74 42 65 51 54 36 56 102 57
62 64 76 67 79 61 81 77 59 38
84 68 71 94 71 86 106 75 91 55
48 82 83 54 79 62 68 58 41 47

a) Construct a stem –and – leaf display for these data. [4]
b) Construct a frequency distribution and relative frequency distribution for these data, using 10 interval width. [6]
c) Construct a cumulative frequency and a cumulative relative frequency distribution. [6]
d) What proportion of the grades is less than 60? [2]
e) What proportion of the grades is more than 70? [2]
QUESTION 3 [20]
The pulse rate of 19 randomly chosen students is measured. Nine of them are smokers and ten non-smokers.
Smokers 62 61 60 65 60 59 58 59 60 62
Non Smokers 74 80 72 79 77 74 78 77 75 75

a) Determine the mean, the median and the mode of the scores for the pulse rate of smokers. [3]
b) Determine the mean, the median and the mode of the scores for the pulse rate of non-smokers [3]
c) Compute the sample variance and the sample standard deviation for the pulse rate of smokers [4]
d) Compute the sample variance and the sample standard deviation for the pulse rate of non-smokers.[4]
e) Compute the range and coefficient of variation for the pulse rate of smokers. [3]
f) Compute the range and coefficient of variation of the pulse rate of non-smokers [3]

QUESTION 4 [20]
The following data represent the age in years of sample of 24 employees from a public service department
30 25 26 23 49 22 33 61 44 28 58 35 41 34 40 44 51 42 48 39 53 27 36 36
a) Find the lower quartile of the ages [5]
b) Find the upper quartile of the ages [5]
c) Find the 60th percentile [5]
d) Compute interquartile range [5]

QUESTION 5 [20]
Given the following data
x 420 610 625 500 400 500 500 650 450 550
y 2.80 3.60 3.75 3.00 2.50 2.70 3.50 3.90 2.95 3.30

a) Calculate the variance and the correlation coefficient [8]
b) Comment on the relationship between x and y [4]
c) Draw the scatter diagram and plot the least squares line. [8]

Answer 1

QUESTION 1

a) Definitions

1. Population of interest: All toys produced by the manufacturer in that production process (i.e., the entire output of children’s toys).
2. Sample: The 500 toys that were drawn and inspected from the production run.
3. Parameter: The true (unknown) long-run proportion of defective toys produced by the manufacturer.
4. Statistic: Any numerical summary computed from the sample; here, the sample defect rate found among the 500 toys.

b) The 3% value is a parameter (it’s the manufacturer’s claim about the true population defect rate).

c) The 5% value is a statistic (it’s computed from the inspected sample of 500 toy QUESTION 2

Data (40 exam grades): shown in prompt.

a) Stem-and-leaf display (stem = tens, leaf = ones 3 | 6 8, 4 | 1 2 7 8, 5 | 1 4 4 5 6 7 8 9 , 6 | 1 2 2 3 4 5 7 8 8 , 7 | 1 1 4 5 6 7 9 9 , 8 | 1 2 3 4 6 , 9 | 1 4 , 10 | 2 6

b) Frequency & relative frequency (class width = 10)

Classes used: [30–39), [40–49), [50–59), [60–69), [70–79), [80–89), [90–99), [100–109).

I’ve placed full tables in your viewer:

1. “Q2 – Frequency & Relative Frequency (width 10)” (counts + proportions)
2. “Q2 – Cumulative Frequency Tables” (cumulative counts + cumulative proportions)

c) Cumulative frequency & cumulative relative frequency

Included in “Q2 – Cumulative Frequency Tables.”

d) Proportion of grades < 60: 0.350

e) Proportion of grades > 70: 0.425

QUESTION 3

Pulse rates (10 smokers, 10 non-smokers).

a) Smokers – mean, median, mode

1. Mean: 60.6
2. Median: 60
3. Mode(s): 60

b) Non-smokers – mean, median, mode

1. Mean: 76.1
2. Median: 76
3. Mode(s): 74, 75, 77 (multimodal)

c) Smokers – sample variance & sample standard deviation

1. Sample variance (n–1): 4.044
2. Sample standard deviation: 2.011

d) Non-smokers – sample variance & sample standard deviation

1. Sample variance (n–1): 6.322
2. Sample standard deviation: 2.514

e) Smokers – range & coefficient of variation

1. Range: 7
2. Coefficient of variation (s/mean): 0.033

f) Non-smokers – range & coefficient of variation

1. Range: 8
2. Coefficient of variation (s/mean): 0.033

(See “Q3 – Pulse Rate Summary” table for a compact view.)

QUESTION 4

Ages (n = 24).

Using standard percentile (linear) definitions:

1. Lower quartile (Q1, 25th percentile): 29.5
2. Upper quartile (Q3, 75th percentile): 45.0
3. 60th percentile (P60): 40.8
4. Interquartile range (IQR = Q3 – Q1): 15.5

(Shown in “Q4 – Quartiles & Percentile” table.)

QUESTION 5

Given xxx and yyy:

a) Sample variances & correlation coefficient

1. Var(xxx) (n–1): 7491.3889
2. Var(yyy) (n–1): 0.2283
3. Pearson correlation rrr: 0.9060

b) Comment

1. There is a strong positive linear relationship between xxx and yyy (r ≈ 0.91). As xxx increases, yyy tends to increase in a near-linear manner.

c) Scatter diagram & least-squares line

1. Least-squares line: y^=0.005002 x+0.596439\; \hat{y} = 0.005002\,x + 0.596439y^​=0.005002x+0.596439
2. I’ve plotted the scatter with the regression line (see the figure in your viewer titled “Q5: Scatter Plot with Least Squares Line.”)