5.8.4 Practice set: statistical inference

Estimators

Estimators: bias, variance and consistency (lesson 5.8.1)

0 of 8 correct

1.
An estimator takes the values 4, 6, 5 and 9, each with probability $\frac{1}{4}$ , and the true value is 5. Find its bias
2.
For the same estimator, find its mean squared error
3.
Draws have standard deviation $\sigma = 3$ . Find the variance of the mean of 25 of them
4.
Find $s^2$ for the sample 2, 4, 6, 8, dividing by $n - 1$
5.
Find the divide-by- $n$ estimate $\hat{\sigma}^2$ for the same sample
6.
Draws have variance 4. Find the mean of the divide-by- $n$ estimate from samples of 10
7.
The mean is known to be $\mu = 4$ . Find the unbiased estimate of $\sigma^2$ from the sample 2, 4, 6, 8
8.
An estimator has bias 0.1 for every $n$ and variance $\frac{1}{n}$ . Find the limit of its mean squared error as $n$ grows

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Sampling distributions and confidence intervals

Sampling distributions and confidence intervals (lesson 5.8.2)

0 of 8 correct

1.
Draws have $\sigma = 0.2$ . Find the standard error of the mean of 100 draws
2.
With $\sigma = 1.5$ known, 36 draws have $\bar{x} = 3.7$ . Find the lower end of the 95% interval for $\mu$
3.
With $\sigma = 2$ known and 16 draws, find the margin of the 90% interval, with $z = 1.645$
4.
Ten values have $\bar{x} = 4$ and estimated standard error $\frac{s}{\sqrt{n}} = 0.3$ ; $t_9 = 2.262$ . Find the upper end of the 95% t interval
5.
With $\sigma = 1$ known, how many draws make the 95% margin 0.1? Give the formula's value before rounding up
6.
Eleven normal values have $s^2 = 4$ , and $\chi^2_{0.975} = 20.483$ for 10 degrees of freedom. Find the lower end of the 95% interval for $\sigma^2$
7.
A 95% interval from 50 draws has width 2. About what width do 200 draws give?
8.
From five normal values, the interval $\bar{x} \pm 1.96\,\frac{s}{\sqrt{n}}$ covers $\mu$ with probability 0.8784. Of 100 such intervals, how many do you expect to miss $\mu$ ?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Hypothesis tests

Hypothesis tests (lesson 5.8.3)

0 of 8 correct

1.
With $\sigma = 2$ known, 64 draws have $\bar{x} = 10.5$ . Find the z statistic for $H_0: \mu = 10$
2.
Nine values have $\bar{x} = 5.6$ and $s = 1.2$ . Find the t statistic for $H_0: \mu = 5$
3.
A t test gives $t = 2.1$ , and $P(T \ge 2.1) = 0.024$ for its degrees of freedom. Find the two-sided p-value
4.
Is that result significant at the 1% level? Type 1 for yes, 0 for no
5.
A test has power 0.85 against an alternative. Find the probability $\beta$ of an error of the second kind
6.
The 99% t interval for $\mu$ runs from 1.2 to 3.4. Is $\mu_0 = 1$ rejected at the 1% level? Type 1 for yes, 0 for no
7.
Ten independent tests of true null hypotheses are run at $\alpha = 0.05$ . How many false rejections do you expect?
8.
A one-sided test with $H_1: \mu > \mu_0$ gives $t = -1.2$ . Is its p-value above 0.5? Type 1 for yes, 0 for no

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

A t interval and a t test

Write two functions for a sample x (a list or NumPy array), as in lessons 5.8.2 and 5.8.3. Use the sample standard deviation dividing by $n - 1$ and the t distribution with $n - 1$ degrees of freedom (scipy.stats.t).

t_interval(x, level=0.95) returns (low, high), the interval $\bar{x} \pm t\,\frac{s}{\sqrt{n}}$ , with $t$ the $\frac{1 + \text{level}}{2}$ quantile.
t_test(x, mu0=0.0, alternative="two-sided") returns (t_stat, p_value) for $H_0: \mu = \mu_0$ . The alternative is "two-sided", "greater" ( $H_1: \mu > \mu_0$ ) or "less".

Both raise ValueError for fewer than 2 values or when every value is the same, so that the standard error is 0. t_interval also raises it if level is not strictly between 0 and 1, and t_test if alternative is anything else.

The tests use lesson 5.8.2's five measurements and 40 exponential draws, and check your p-values against SciPy's ttest_1samp.

See the tests

import numpy as np
from scipy.stats import ttest_1samp

FIVE = [10.2, 9.6, 10.9, 10.4, 9.9]
EXPO = np.random.default_rng(5829).exponential(1.0, size=40)


def _raises(f, *args, **kwargs):
    try:
        f(*args, **kwargs)
    except ValueError:
        return True
    return False


def test_five_measurements():
    """Lesson 5.8.2's five measurements: the 95% interval runs from about 9.5854 to 10.8146"""
    got = t_interval(FIVE)
    assert got is not None, "t_interval returned None (did you forget to return?)"
    if abs(got[0] - 9.766135505) < 1e-6:
        raise AssertionError("got 9.7661 to 10.6339: that uses 1.96. Use the t quantile with n - 1 degrees of freedom")
    if abs(got[0] - 9.650291915) < 1e-6:
        raise AssertionError("got a lower end of 9.6503: that divides by n. Use the sample standard deviation with ddof=1")
    expect_close(got, (9.585407676840441, 10.814592323159557), what="t_interval(five)", rel=1e-9)


def test_other_level():
    """At 99% the interval widens to about 9.1808 to 11.2192"""
    expect_close(t_interval(FIVE, 0.99), (9.180840155012387, 11.219159844987612), what="t_interval(five, 0.99)", rel=1e-9)


def test_forty_values():
    """Lesson 5.8.2's 40 exponential draws: about 0.6759 to 1.5209"""
    expect_close(t_interval(EXPO), (0.6758823095909889, 1.5208754761807728), what="t_interval(forty draws)", rel=1e-9)


def test_two_sided():
    """Testing 10 gives t about 0.9035 and p about 0.4173; testing 9.5 gives 3.1623 and 0.0341"""
    got = t_test(FIVE, 10)
    assert got is not None, "t_test returned None (did you forget to return?)"
    expect_close(got, (0.9035079029052476, 0.41734905117810633), what="t_test(five, 10)", rel=1e-9)
    expect_close(t_test(FIVE, 9.5), (3.1622776601683746, 0.034109423167409766), what="t_test(five, 9.5)", rel=1e-9)


def test_one_sided():
    """Against 9.5, 'greater' gives p about 0.0171 and 'less' about 0.9829"""
    expect_close(t_test(FIVE, 9.5, "greater"), (3.1622776601683746, 0.017054711583704883), what="t_test(five, 9.5, 'greater')", rel=1e-9)
    expect_close(t_test(FIVE, 9.5, "less"), (3.1622776601683746, 0.9829452884162951), what="t_test(five, 9.5, 'less')", rel=1e-9)


def test_negative_t():
    """A sample below the claimed value gives a negative t and the same two-sided p as SciPy"""
    res = ttest_1samp(EXPO, 1.6)
    expect_close(t_test(EXPO, 1.6), (float(res.statistic), float(res.pvalue)), what="t_test(forty draws, 1.6)", rel=1e-9)


def test_interval_and_test_agree():
    """A claimed value inside the 95% interval has p above 0.05; one outside has p below"""
    low, high = t_interval(FIVE)
    assert t_test(FIVE, (low + high) / 2 + 0.9 * (high - low) / 2)[1] > 0.05, "a value inside the interval should have p > 0.05"
    assert t_test(FIVE, high + 0.01)[1] < 0.05, "a value just outside the interval should have p < 0.05"


def test_bad_inputs():
    """One value, identical values, a level of 0 or 1, or an unknown alternative raise ValueError"""
    assert _raises(t_interval, [3.0]), "t_interval with one value should raise ValueError"
    assert _raises(t_test, [3.0]), "t_test with one value should raise ValueError"
    assert _raises(t_interval, [2.0, 2.0, 2.0]), "t_interval with identical values should raise ValueError"
    assert _raises(t_test, [2.0, 2.0, 2.0], 1.0), "t_test with identical values should raise ValueError"
    for level in (0, 1):
        assert _raises(t_interval, FIVE, level), f"t_interval(level={level}) should raise ValueError"
    assert _raises(t_test, FIVE, 10, "bigger"), "t_test with alternative='bigger' should raise ValueError"

Estimators

Sampling distributions and confidence intervals

Hypothesis tests

Coding exercise

Free further reading