Statistical inference – university Year 1 maths worksheet with answers to download as PDF

A random sample of 8 values is: 12, 15, 11, 18, 14, 16, 13, 17.

1. Find the sample mean \(\bar x\). [1 mark]
2. Find the sample variance \(s^2=\dfrac1{n-1}\sum(x_i-\bar x)^2\) and the standard deviation \(s\). [3 marks]
3. Find the estimated standard error of the mean. [2 marks]
4. Explain why we divide by \(n-1\) and not by \(n\). [2 marks]

A population has mean \(\mu=40\) and standard deviation \(\sigma=10\). Samples of size \(n=25\) are taken.

1. State the approximate distribution of \(\bar X\) and justify it. [2 marks]
2. Find \(P(\bar X>43)\). [2 marks]
3. Find \(P(38\lt \bar X<42)\). [2 marks]
4. How large must \(n\) be for the standard error to be at most 1? [2 marks]

Find a 95% confidence interval in each case.

1. \(\sigma=6\) known, \(n=36\), \(\bar x=72.4\). [2 marks]
2. A normal population, \(n=10\), \(\bar x=15.2\), \(s=2.1\). [3 marks]
3. 84 successes in a sample of 200 (a proportion). [3 marks]

Carry out each test, stating the hypotheses and the conclusion.

1. \(H_0:\mu=500\) against \(H_1:\mu\ne500\) with \(\sigma=20\), \(n=16\), \(\bar x=509\), at the 5% level. [3 marks]
2. The same data with \(H_1:\mu>500\). [2 marks]
3. A normal sample with \(n=12\), \(\bar x=4.6\), \(s=0.9\): test \(H_0:\mu=4\) against \(H_1:\mu\ne4\) at 5%. [3 marks]

A die is rolled 120 times.

1. State the hypotheses and the expected frequencies. [2 marks]
2. Calculate \(\chi^2=\sum\dfrac{(O-E)^2}E\). [3 marks]
3. Test at the 5% level. [2 marks]

Five pairs of data are given.

1. Find \(S_{xx}\), \(S_{xy}\) and \(S_{yy}\). [3 marks]
2. Find the least-squares line \(y=a+bx\). [2 marks]
3. Find the correlation coefficient \(r\) and comment. [2 marks]

A consumer magazine tests 40 phones of a new model. The mean battery life is \(\bar x=18.6\) hours with standard deviation \(s=2.5\) hours. The manufacturer advertises "20 hours of battery".

1. Find a 95% confidence interval for the mean battery life.
2. Does the interval support the advert? Explain.
3. Find a 99% interval and explain why it is wider.
4. How many phones should be tested to estimate the mean to within \(\pm0.5\) hours with 95% confidence?

A café's coffee machine is set to pour 250 ml. The volume poured is normal with \(\sigma=8\) ml. The manager measures 25 cups and finds a mean of \(246.5\) ml.

1. State suitable hypotheses.
2. Calculate the test statistic and the \(p\)-value.
3. Give conclusions at the 5% and at the 1% levels.
4. Explain in context what a Type I error would be.

In a random sample of 1 000 voters, 520 say they will vote for the Green candidate. A newspaper headline says "Green candidate set to win a majority".

1. Find a 95% confidence interval for the proportion \(p\) of supporters.
2. Test \(H_0:p=0.5\) against \(H_1:p>0.5\) at the 5% level.
3. Is the headline justified?
4. What sample size would give a margin of error of 2 percentage points?

Eight students take a test, use a revision app for two weeks, then take a similar test. The improvements \(d=\) after \(-\) before are shown.

1. Explain why a paired test is appropriate.
2. Find \(\bar d\) and \(s_d\).
3. Test at the 5% level whether the mean improvement is positive.
4. Suggest a weakness of this study design.

Two production lines make the same component. Samples give the following lengths (in mm).

1. State hypotheses for a test of whether the mean lengths differ.
2. Calculate the standard error of \(\bar X_1-\bar X_2\).
3. Calculate the test statistic and the \(p\)-value.
4. State your conclusion at the 5% and 10% levels.

A council records 130 road accidents on weekdays over a year.

1. State hypotheses for a test of whether accidents are equally likely on each weekday.
2. Calculate the expected frequencies and \(\chi^2\).
3. Carry out the test at the 5% level.
4. Which day contributes most to \(\chi^2\), and what might explain it?

An ice cream van records the midday temperature and the number of ice creams sold on six summer days.

1. Given \(S_{xx}=167.33\) and \(S_{xy}=1586.67\), find the regression line of \(y\) on \(x\).
2. Interpret the gradient in context.
3. Predict the sales at 25 \(^\circ\)C.
4. Given \(S_{yy}=15120.8\), find \(r\).
5. Would you use the line to predict sales at 5 \(^\circ\)C? Explain.

Bags of flour should contain 100 g on average; the mass is normal with \(\sigma=4\) g. An inspector weighs 16 bags and tests \(H_0:\mu=100\) against \(H_1:\mu<100\) at the 5% level.

1. Show that \(H_0\) is rejected when \(\bar x<98.36\).
2. What is the probability of a Type I error?
3. If in fact \(\mu=97\), find the probability of a Type II error.
4. Find the power of the test when \(\mu=97\) and explain how it could be increased.

A health survey of 170 adults records their level of exercise and the quality of their sleep.

1. State the hypotheses for a test of independence.
2. Show that the expected frequency for "good sleep, low exercise" is \(29.41\) and find the others.
3. Calculate \(\chi^2\).
4. Find the degrees of freedom and test at the 1% level.
5. A journalist writes "exercise improves sleep". Comment.

A single fair die has mean \(\mu=3.5\) and variance \(\sigma^2=\dfrac{35}{12}\). A class repeatedly rolls 30 dice and records the mean score \(\bar X\). Out of 200 samples, the results were:

1. Show that \(\sigma^2=\dfrac{35}{12}\).
2. State the approximate distribution of \(\bar X\).
3. Find \(P(\bar X>4)\) and the expected number of samples above 4.
4. Compare with the class results.
5. Explain why the distribution of \(\bar X\) is close to normal although a single die is uniform.

An engineer tests six steel cables for a footbridge. The breaking loads (in kN) are below. The specification requires a mean breaking load above 50 kN. Assume the loads are normally distributed.

1. Calculate \(\bar x\) and \(s\).
2. Find a 95% confidence interval for the mean breaking load.
3. Test \(H_0:\mu=50\) against \(H_1:\mu>50\) at the 5% level.
4. The engineer says "the cables are within specification". Is this justified? What would you recommend?

Over 10 summer weeks at a seaside resort, the number of ice creams sold and the number of lifeguard rescues give a sample correlation coefficient \(r=0.72\).

1. To test \(H_0:\rho=0\) the statistic \(t=r\sqrt{\dfrac{n-2}{1-r^2}}\) has a \(t_{n-2}\) distribution. Calculate it.
2. Test at the 5% level (two-sided).
3. A local councillor suggests banning ice cream to reduce rescues. Explain the flaw.
4. Name the likely lurking variable and describe a better analysis.

Researchers plan a study to detect whether a new diet lowers blood pressure by 2 mmHg on average. Individual changes have standard deviation \(\sigma=8\) mmHg. They will test \(H_0:\mu=0\) against \(H_1:\mu\ne0\) at the 5% level.

1. For a sample of size \(n\), write the standard error and the rejection rule.
2. For \(n=100\), find the power of the test when \(\mu=2\) (ignore the very small chance of rejecting in the wrong tail).
3. Show that a power of 90% needs \(n\ge\left(\dfrac{(1.96+1.2816)\times8}{2}\right)^2\) and evaluate it.
4. Explain the trade-off between the significance level, the power and the cost of the trial.

Let \(X_1,\dots,X_n\) be independent with mean \(\mu\) and variance \(\sigma^2\), and let \(\bar X=\frac1n\sum X_i\).

1. Show that \(E[\bar X]=\mu\) and \(\operatorname{Var}(\bar X)=\dfrac{\sigma^2}n\).
2. Show that \(\sum(X_i-\bar X)^2=\sum X_i^2-n\bar X^2\).
3. Using \(E[X_i^2]=\sigma^2+\mu^2\) and \(E[\bar X^2]=\dfrac{\sigma^2}n+\mu^2\), show that \(E\left[\sum(X_i-\bar X)^2\right]=(n-1)\sigma^2\).
4. Deduce that \(S^2=\dfrac1{n-1}\sum(X_i-\bar X)^2\) is unbiased, and find the bias of the version with \(\frac1n\).