How to promote and advertise your Business

 Great! Here are clear, exam-ready answers in simple PGDE style.

(For Q3 the table had one missing frequency. I assumed 5 for 64–72 to keep the distribution symmetric. I show all steps.)

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1) Six roles of statistics in education

• Planning: guides lesson/exam design using past data.

• Measurement: helps score, grade, and summarize results.

• Evaluation: compares teaching methods and curricula.

• Decision-making: supports school policies with evidence.

• Research: enables data collection, analysis, and inference.

• Prediction: forecasts enrolment/performance needs.

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2) Measurement scales (with uses)

• Nominal: names/groups only (no order). Example: gender; school house.

• Ordinal: ordered ranks (unequal gaps). Example: 1st–2nd–3rd position.

• Interval: equal gaps, no true zero. Example: Celsius temperature.

• Ratio: equal gaps, true zero. Example: test scores, height, weight.

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3) Grouped distribution & Z/Raw conversions

Assumed frequencies:
10–18:2, 19–27:3, 28–36:5, 37–45:9, 46–54:12, 55–63:9, 64–72:5, 73–81:3, 82–90:2.

• Midpoints (X): 14, 23, 32, 41, 50, 59, 68, 77, 86

• Total $N=50$

• $\bar X=\dfrac{\sum fX}{\sum f}=50.0$

• $SD=\sqrt{\dfrac{\sum fX^2}{\sum f}-\bar X^2}\approx 16.89$

(a) Convert raw to Z

$$Z=\frac{X-\bar X}{SD}$$

• For 70%: $Z=\frac{70-50}{16.89}\approx \mathbf{1.18}$

• For 35%: $Z=\frac{35-50}{16.89}\approx \mathbf{-0.89}$

(b) Convert Z to raw

$$X=Z\cdot SD+\bar X$$

• For Z=2: $X=2(16.89)+50\approx \mathbf{83.77}$

• For Z=-1.5: $X=-1.5(16.89)+50\approx \mathbf{24.67}$

(If your missing frequency is different, recompute mean/SD, then apply the same formulas.)

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4) (a) Degree of Freedom (df)

Number of values free to vary after fixing a constraint.
For a sample of size $n$ with a known mean, $df=n-1$.

(b) Three uses of Standard Deviation

• Measures how spread out scores are.

• Compares variability between groups.

• Used to compute Z-scores and confidence intervals.

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5) (a) Mean by coding

Data: 15, 18, 21, 24, 27, 30, 33, 36, 42
Let assumed mean $A=27$, class width $c=3$.
Codes $u=(X-A)/c = -4,-3,-2,-1,0,1,2,3,5$; $\sum u=1$, $n=9$.

$$\bar X=A+\frac{\sum u}{n}c=27+\frac{1}{9}\cdot 3= \mathbf{27.33}$$

(b) Quota sampling

A non-probability method: divide population into groups (e.g., sex, class) and select participants until each group’s preset quota is filled.

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6) Convert to T-scores ($\bar X=45$, $SD=12$)

$$T=50+10\left(\frac{X-\bar X}{SD}\right)$$

• For 79: $Z=\frac{79-45}{12}=2.83 \Rightarrow T=50+10(2.83)\approx \mathbf{78.3}$

• For 15: $Z=\frac{15-45}{12}=-2.5 \Rightarrow T=50+10(-2.5)=\mathbf{25}$

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