Sampling and Estimation [Video Tutorial]

This tutorial covers material encountered in chapter 17 of the VCE Mathematical Methods Textbook, namely:

• Sampling and estimation methods
• The population, sample and their proportions
• Sampling distributions
• Point estimates
• The 95% confidence interval and the margin of error

Q1 – Population, Sample and their Proportions

Q2 – Point Estimate and 95% Confidence Interval

Q3 – Sampling and Estimation

Q4 – Sample Proportion and Margin of Error

Q5 – Find Probability for Sample Proportions

Q6 – Sample Proportion and Margin of Error 2

Worksheet

Q1. Triassic Park has 200 dinosaurs, 80 of them carnivores. A random sample of 30 dinosaurs was selected, and 14 of them were carnivores. In this example:

(a) What’s the population?

(b) What’s the population proportion?

(c) What’s the sample proportion? Do you think the random sample is representative of the entire population?

Q2. A standard six-sided die is cast 50 times, and \(n\) even numbers were observed.

(a) Give a point estimate for \(p\), the probability of observing an even number when the die is cast.

(b) Give an expression for a 95% confidence interval for \(p\).

Q3. To study the effectiveness of video games for reducing stress levels, a researcher measured the stress levels of 100 people who regularly play League of Legends and measured their stress levels at the end of a 4 hour gaming session.

(a) Do you think this sample is representative of the general population? Give a general explanation.

(b) How would you suggest the sample be chosen?

(c) Bonus: Do you think the subjects’ stress levels would generally decrease or increase?

Q4. A sample of \(k\) people were asked if they liked mangos, with 85% of people saying yes.

(a) Find the value of the sample proportion \(\hat{p}\).

(b) Give an expression for the margin of error, \(M\), for this estimate at the 95% confidence level in terms of \(k\).

(c) If the number of people in the sample were tripled, what would be the effect on the margin of error \(M\)?

Q5. Suppose that 30 independent random samples were taken from a large population, and that a 95% confidence interval for the population proportion \(p\) was computed from each of these samples.

(a) How many of the 95% confidence intervals would you expect to contain the population proportion \(p\)?

(b) Give an expression for the probability that all 30 confidence intervals contain \(p\).

Q6. Tom has determined that an approximate 95% confidence interval for the proportion of people subscribed to his blog who regularly read new entries is (0.60, 0.80).

(a) What \(\hat{p}\) value was used to determine this confidence interval?

(b) Write down an expression for the margin of error \(M\).

(c) Comment on how Tom could increase the precision of his findings.

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