Crowds behave better than individuals

Mathematics · Data & Statistics · ages 17-18

Name ______________________ Date ____________

Independent observations come from a population with standard deviation 12. For samples of size 36, what is the standard error of the mean?
Answer: ______________
Each sample contains 4 independent observations from the same population with finite positive standard deviation. What is the ratio of the sample mean's standard error to that population standard deviation? Give a decimal.
Answer: ______________
Two populations have standard deviations 18 and 24. Each sample contains 36 independent observations from its population. Which population gives the larger standard error of the sample mean?
- The population with σ = 18
- The population with σ = 24
- Both give the same spread
- Their means decide the spread
For independent random observations from the same population with finite, positive variance, what happens to the standard error of the mean as sample size grows?
- The standard error decreases
- The standard error increases
- The standard error is unchanged
Take 16 independent observations from a normal population with mean 65 and standard deviation 16. A left-cumulative table gives Φ(1) = 0.8413. Using that entry, find the probability that the sample mean exceeds 69.
Answer: ______________
Independent observations come from one population with finite mean and finite positive variance. When n is large enough for the central limit theorem approximation, what shape does the distribution of sample means have?
- Uniform over a fixed interval
- Strongly skewed in every case
- Approximately normal
Independent observations come from a normal population. What can be said about the sample mean for samples of size 4?
- It is normal only when n reaches 30
- It must have the population's spread
- It is exactly normally distributed
- It must be uniform rather than normal
Dan takes 64 independent observations from a population with standard deviation 24. He reports 24 / 64 = 0.375 as the standard error. What is the correct standard error divided by his reported value?
Answer: ______________
For independent observations from a population with standard deviation 20, samples of 25 have standard error 4. What standard error do samples of 100 have?
Answer: ______________
For independent random observations from one population with finite mean and variance, increasing sample size moves the expected sample mean towards the population mean.
Circle one: True False

Answer key

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Crowds behave better than individuals W1-mt_nugaEeUSii-s1

2 · Standard error = 12 / √36 = 12 / 6 = 2.
0.5 · The ratio is 1 / √n. With n = 4, it is 1 / 2 = 0.5.
The population with σ = 24 · Both denominators are √36 = 6. The standard errors are 18 / 6 = 3 and 24 / 6 = 4, so the population with σ = 24 gives the larger one.
The standard error decreases · The standard error is σ / √n. The population's σ stays fixed while √n grows, so the quotient decreases.
0.1587 · The standard error is 16 / √16 = 4. The boundary has z = (69 - 65) / 4 = 1. The upper-tail probability is approximately 1 - 0.8413 = 0.1587.
Approximately normal · The central limit theorem supports an approximately normal sampling distribution for sufficiently large n under the stated assumptions.
It is exactly normally distributed · Independent observations from a normal population have an exactly normal sample mean for every positive n. Its standard error is σ / √4 = σ / 2.
8 · Dan divides by n instead of √n. The correct standard error is 24 / √64 = 24 / 8 = 3. Dividing it by his reported value gives 3 / 0.375 = 8.
2 · Standard error = 20 / √100 = 20 / 10 = 2. Multiplying sample size by four halves the standard error.
False · The expected sample mean already equals μ for every sample size under these assumptions. Increasing n reduces its standard error rather than moving its centre.