Study Notes on Random Samples and Proportions

Overview of Random Samples and Proportions

Concept of Random Samples

• Random samples help estimate the population characteristics by selecting a number of observations from the population.

• In this scenario, random samples of 100 people each are used to determine the support level for a local basketball team in comparison to others in the league.

Sample Proportions Data

• The following sample proportions from each of the 10 samples were documented:

  • Sample 1: 0.80

  • Sample 2: 0.75

  • Sample 3: 0.69

  • Sample 4: 0.40

  • Sample 5: 0.98

  • Sample 6: 0.92

  • Sample 7: 0.83

  • Sample 8: 0.76

  • Sample 9: 0.87

  • Sample 10: 0.93

Interpretation of Proportions

Question a: Meaning of the Proportion 0.98 for the 5th Sample

• The proportion 0.98 in the 5th sample indicates that 98% of the 100 people surveyed in that specific sample support the local basketball team.

• This high level of support could reflect positive sentiments towards the team, but it may not represent broader public opinion due to the sample size being only from 100 individuals.

Question b: Margin of Error for Population Proportion Estimate

• The approximate mean of the sample proportions is 0.79, and the standard deviation is 0.16.

• To determine the margin of error (ME) for estimating the population proportion, the following formula can be used:

  • $ME = z \times \left(\frac{s}{\sqrt{n}}\right)$

  • where:

  • $z$ is the z-score corresponding to the desired confidence level (e.g., 1.96 for a 95% confidence level).

  • $s$ is the standard deviation of the sample proportions.

  • $n$ is the sample size, which in this case is 100.

• Therefore, the calculation of the margin of error can be performed using the aforementioned formula.

  • If we assume a z-score of 1.96 for a 95% confidence interval:

  • $ME = 1.96 \times \left(\frac{0.16}{\sqrt{100}}\right)$

  • $ME = 1.96 \times \left(\frac{0.16}{10}\right)$

  • $ME = 1.96 \times 0.016$

  • $ME = 0.03136$

• Therefore, the margin of error for the estimate of the population proportion is approximately 0.0314 or 3.14%. This means we can expect the true population proportion to fall within the interval defined by the sample proportion ± 3.14%.