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9709/72

Mathematics 9709/72 — February/March 2017

Cambridge A-Level · Probability & Statistics 2 · worked solutions for every part, with the mark scheme

7
questions
50
marks
75
minutes

Topics Sampling and Estimation · Hypothesis Tests · The Poisson Distribution · Linear Combinations of Random Variables · Continuous Random Variables

Q14MSampling and EstimationFree sample

In a survey, 36 out of 120 randomly selected voters in Hungton said that if there were an election next week they would vote for the Alpha party. Calculate an approximate 90% confidence interval for the proportion of voters in Hungton who would vote for the Alpha party.

DifficultyMedium-Easy
Worked solution

Approach

The sample proportion is p^=36120=0.3\hat{p} = \frac{36}{120} = 0.3. An approximate confidence interval for a population proportion is given by p^±zp^(1−p^)n\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}, where zz is the standard normal value for the required confidence level. For a 90% confidence interval, z=1.645z = 1.645.

Working

Calculate the sample proportion:

p^=36120=0.3\hat{p} = \frac{36}{120} = 0.3

Calculate the variance of the sample proportion:

Var(p^)=p^(1−p^)n=0.3(1−0.3)120=0.3×0.7120=0.21120=0.00175\text{Var}(\hat{p}) = \frac{\hat{p}(1-\hat{p})}{n} = \frac{0.3(1-0.3)}{120} = \frac{0.3 \times 0.7}{120} = \frac{0.21}{120} = 0.00175

The standard error is:

0.00175≈0.04183\sqrt{0.00175} \approx 0.04183

For 90% confidence, z=1.645z = 1.645. The confidence interval is:

0.3±1.6450.001750.3 \pm 1.645\sqrt{0.00175} =0.3±1.645(0.04183)= 0.3 \pm 1.645(0.04183) =0.3±0.06882= 0.3 \pm 0.06882

Lower limit:

0.3−0.06882=0.23118≈0.2310.3 - 0.06882 = 0.23118 \approx 0.231

Upper limit:

0.3+0.06882=0.36882≈0.3690.3 + 0.06882 = 0.36882 \approx 0.369

Answer

The approximate 90% confidence interval is:

0.231 to 0.369(3 sf)0.231 \text{ to } 0.369 \quad (3 \text{ sf})
Final answer

0.231 to 0.369 (3 sf)

Detailed explanation

Walkthrough

First, find the sample proportion: 36 out of 120 voters said they would vote for the Alpha party, so the sample proportion is p^=36120=0.3\hat{p} = \frac{36}{120} = 0.3. This is our point estimate of the population proportion.

Next, we need the variance of the sample proportion. For a proportion, the variance of the sample proportion is given by p^(1−p^)n\frac{\hat{p}(1-\hat{p})}{n}. This formula comes from the fact that, for large samples, the sample proportion is approximately normally distributed with mean pp and variance p(1−p)n\frac{p(1-p)}{n}. We estimate pp using p^\hat{p}, so:

Var(p^)=0.3(1−0.3)120=0.3×0.7120=0.21120=0.00175\text{Var}(\hat{p}) = \frac{0.3(1-0.3)}{120} = \frac{0.3 \times 0.7}{120} = \frac{0.21}{120} = 0.00175

The standard error is the square root of the variance:

0.00175≈0.04183\sqrt{0.00175} \approx 0.04183

For a 90% confidence interval, we use the zz-value that leaves 5% in each tail of the standard normal distribution: z=1.645z = 1.645. (This is different from 1.96, which is used for 95% confidence.)

The confidence interval is:

p^±z×SE=0.3±1.645×0.04183=0.3±0.06882\hat{p} \pm z \times \text{SE} = 0.3 \pm 1.645 \times 0.04183 = 0.3 \pm 0.06882

This gives a lower limit of 0.3−0.06882=0.2310.3 - 0.06882 = 0.231 and an upper limit of 0.3+0.06882=0.3690.3 + 0.06882 = 0.369.

So we are 90% confident that the true proportion of voters in Hungton who would vote for the Alpha party lies between 0.231 and 0.369.

Key Takeaways

  • The sample proportion p^=xn\hat{p} = \frac{x}{n} is the point estimate of the population proportion.
  • The variance of the sample proportion is p^(1−p^)n\frac{\hat{p}(1-\hat{p})}{n}, and its standard error is the square root of this.
  • For a 90% confidence interval, the zz-value is 1.645 (5% in each tail).
  • The confidence interval formula is p^±zp^(1−p^)n\hat{p} \pm z\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}.

Common Mistakes

  • Using a probability value (like 0.90) instead of the zz-value 1.645. The mark scheme explicitly requires a zz-value, not a probability.
  • Forgetting to divide by nn when calculating the variance of the sample proportion.
  • Using the wrong zz-value, e.g., 1.96 for 95% confidence instead of 1.645 for 90%.
  • Rounding intermediate values too early, which can change the final answer.

Things to Be Careful About

  • Make sure the sample proportion is calculated correctly: 36120=0.3\frac{36}{120} = 0.3.
  • The final answer should be given to 3 significant figures.
  • The interval is written as "lower to upper", e.g., 0.231 to 0.369.
  • The mark scheme requires the zz-value to be used in the interval calculation; using a probability instead loses the mark.
Techniques used
calculate the sample proportioncompute the variance of the sample proportionapply the z-value for 90% confidenceconstruct the confidence interval

The rest of this paper

6 more questions
  • Q2Hypothesis Tests4M
  • Q3Sampling and Estimation5M
  • Q4Hypothesis Tests · The Poisson Distribution · Linear Combinations of Random Variables7M
  • Q5Continuous Random Variables9M
  • Q6Linear Combinations of Random Variables10M
  • Q7The Poisson Distribution11M
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