When must cells be combined?

When an expected frequency Ei<5: merge with a neighbouring cell, adding both O and E.

Degrees of freedom for a goodness of fit test?

Cells (after combining) -1- number of parameters estimated from the data.

Degrees of freedom for a fair-die test?

6-1=5, since nothing is estimated.

Expected frequency in a contingency table?

\frac{row total\timescolumn total}{grand total}

Hypotheses for a contingency table test?

H0: no association (the variables are independent); H1: there is an association.

How do you estimate p for a B(N,p) fit?

\hat p=\frac{\bar x}{N}, using the sample mean. This costs one degree of freedom.

How do you estimate \lambda for a Poisson fit?

\hat\lambda=\bar x, the sample mean. This costs one degree of freedom.

When do you reject the null hypothesis?

When X^2\geq the critical value, or the p-value is less than the significance level.

Goodness of fit tests and contingency tablesEdexcel A-Level Further Maths: Flashcards

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State the $\chi^2$ test statistic.

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State the $\chi^2$ test statistic.: $X^2=\sum\frac{(O_i-E_i)^2}{E_i}$
When must cells be combined?: When an expected frequency $E_i<5$ : merge with a neighbouring cell, adding both $O$ and $E$ .
Degrees of freedom for a goodness of fit test?: Cells (after combining) $-1-$ number of parameters estimated from the data.
Degrees of freedom for a fair-die test?: $6-1=5$ , since nothing is estimated.
Expected frequency in a contingency table?: $\frac{\text{row total}\times\text{column total}}{\text{grand total}}$
Degrees of freedom for a contingency table?: $(r-1)(c-1)$
Hypotheses for a contingency table test?: $\mathrm{H}_0$ : no association (the variables are independent); $\mathrm{H}_1$ : there is an association.
How do you estimate $p$ for a $\mathrm{B}(N,p)$ fit?: $\hat p=\frac{\bar x}{N}$ , using the sample mean. This costs one degree of freedom.
How do you estimate $\lambda$ for a Poisson fit?: $\hat\lambda=\bar x$ , the sample mean. This costs one degree of freedom.
Expected frequency for each outcome of a discrete uniform model with $k$ outcomes and $n$ observations?: $\frac nk$
Critical value of $\chi^2_5$ at 5%?: $11.070$
Critical value of $\chi^2_2$ at 5%?: $5.991$
When do you reject the null hypothesis?: When $X^2\geq$ the critical value, or the p-value is less than the significance level.

Exam questions on Goodness of fit tests and contingency tables

A six-sided die is rolled 120 times. The frequencies of the faces $1,2,3,4,5,6$ are $14, 25, 17, 22, 24, 18$ respectively. A test is carried out, at the 5% significance level, of whether the die is fair.
Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$ .2 marks
A survey of 120 students recorded their gender and usual mode of travel to school. Of the 60 boys, 20 walk, 25 take the bus and 15 come by car. Of the 60 girls, 30 walk, 20 take the bus and 10 come by car. A $\chi^2$ test is used to investigate whether mode of travel is associated with gender.
The test statistic is $3.56$ , to 3 significant figures. State the conclusion of the test at the 5% significance level, giving the critical value used.2 marks
A farmer packs eggs in boxes of 4. In a random sample of 100 boxes, the numbers of boxes containing $0,1,2,3,4$ cracked eggs were $41, 38, 16, 4, 1$ respectively. The farmer suggests that the number of cracked eggs in a box follows a binomial distribution $\mathrm{B}(4,p)$ , where $p$ is estimated from the data.
Show that the estimate of $p$ is $0.215$ , and find the expected frequency of boxes containing exactly one cracked egg, to 2 decimal places.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).