Practice: Sampling and statistical reasoning

Sampling and statistical reasoning

Warm-upHuman

A news website asks its readers to click “yes” or “no”: “Should the city ban cars from the old town on Sundays?” $12\,000$ readers vote, and $71\%$ say no. What is the main problem with using this to estimate the opinion of all the city's residents?

Choose one

Voluntary response: readers choose whether to vote, so people with strong feelings — and readers of that website — are over-represented.The sample is too small:

12\,000

is a tiny fraction of the city.Nothing: with

12\,000

responses the margin of error is under

1

percentage point.The question has only two answers.

Sampling and statistical reasoning

Warm-upHuman

To estimate the proportion of households in a region that own a bicycle, which survey would you trust more?

Choose one

A simple random sample of

800

households.

20\,000

households who answered a questionnaire handed out at bicycle shops.They are equally good, since both are large.

Sampling and statistical reasoning

CoreHuman

A random survey of $500$ people has a margin of error of about $4.4$ percentage points. By what factor must the sample size be multiplied to make the margin of error about half as large?

Sampling and statistical reasoning

CoreHuman

From a random sample, a researcher computes a $95\%$ confidence interval for the proportion of adults in a country who use mobile banking: $0.52$ to $0.58$ . Which interpretation is correct (in the usual, frequentist sense)?

Choose one

The method used produces intervals that contain the true proportion for about

95\%

of random samples; this interval was produced by that method.There is a

95\%

probability that the true proportion is between

0.52

and

0.58

.

95\%

of adults use mobile banking at a rate between

52\%

and

58\%

.If the survey were repeated,

95\%

of the new estimates would fall between

0.52

and

0.58

.

Sampling and statistical reasoning

CoreHuman

Fit the least-squares line to the five points $(1, 2), (2, 2), (3, 5), (4, 6), (5, 10)$ . What value of $y$ does the line predict at $x = 6$ ?

Sampling and statistical reasoning

CoreHuman

For a data set, the correlation coefficient is $r = 0$ . What can you conclude?

Choose one

There is no linear association: the least-squares line is horizontal (

\hat b = 0

). There may still be a strong non-linear relationship.

x

and

y

are unrelated.The points lie on a horizontal line.Half the points lie above the line

y = \bar y

.

Sampling and statistical reasoning

CoreHuman

Across $50$ countries, the number of mobile phones per person is strongly positively correlated with life expectancy. Which conclusion is best supported?

Choose one

The correlation may be explained by a confounder, such as national wealth, which tends to increase both; it does not by itself show that phones lengthen lives.Giving people mobile phones will increase life expectancy.Living longer causes people to buy more phones.Because the correlation is strong, it must be causal.

Sampling and statistical reasoning

StretchHuman

In the proof that $\sum_i (x_i - c)^2 = \sum_i (x_i - \bar x)^2 + n(\bar x - c)^2$ , the expansion produces a cross term $2(\bar x - c)\sum_i (x_i - \bar x)$ . Why is it zero?

Choose one

Because

\sum_i (x_i - \bar x) = \sum_i x_i - n\bar x = 0

, by the definition of the mean.Because

c = \bar x

.Because each

x_i - \bar x

is zero.Because squares are never negative.

Sampling and statistical reasoning

StretchFormal

Data are x : Fin n → ℝ and mean x is their mean. Which Lean statement says that the mean minimises the sum of squared deviations?

Choose one

∀ c : ℝ, ∑ i, (x i - mean x) ^ 2 ≤ ∑ i, (x i - c) ^ 2∀ c : ℝ, ∑ i, (x i - c) ^ 2 ≤ ∑ i, (x i - mean x) ^ 2∃ c : ℝ, ∑ i, (x i - mean x) ^ 2 ≤ ∑ i, (x i - c) ^ 2∀ c : ℝ, ∑ i, |x i - mean x| ≤ ∑ i, |x i - c|

Sampling and statistical reasoning

StretchComputational

Write fit(xs, ys) returning the pair (a, b) of the least-squares line $y = a + bx$ , and correlation(xs, ys) returning $r$ , using $S_{xx}$ , $S_{xy}$ and $S_{yy}$ as in the lesson.

Runs entirely in your browser, in an isolated worker. Nothing is sent to a server.