| Exam Board | OCR MEI |
|---|---|
| Module | S3 (Statistics 3) |
| Year | 2010 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chi-squared goodness of fit |
| Type | Chi-squared goodness of fit: Binomial |
| Difficulty | Standard +0.3 This is a standard chi-squared goodness of fit test with straightforward calculations. Part (i) requires computing expected frequencies for B(3,0.5) and performing a routine test. Part (ii) involves basic mean calculation and parameter estimation. Part (iii) is given the test statistic, requiring only comparison with critical values. Part (iv) asks for conceptual understanding but at a basic level. While it's a multi-part question worth several marks, each component follows textbook procedures with no novel insight required, making it slightly easier than average for an S3 question. |
| Spec | 5.06b Fit prescribed distribution: chi-squared test |
| Number of eggs hatched | 0 | 1 | 2 | 3 |
| Number of nests | 7 | 23 | 29 | 21 |
1 Coastal wildlife wardens are monitoring populations of herring gulls. Herring gulls usually lay 3 eggs per nest and the wardens wish to model the number of eggs per nest that hatch. They assume that the situation can be modelled by the binomial distribution $\mathrm { B } ( 3 , p )$ where $p$ is the probability that an egg hatches. A random sample of 80 nests each containing 3 eggs has been observed with the following results.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | }
\hline
Number of eggs hatched & 0 & 1 & 2 & 3 \\
\hline
Number of nests & 7 & 23 & 29 & 21 \\
\hline
\end{tabular}
\end{center}
(i) Initially it is assumed that the value of $p$ is $\frac { 1 } { 2 }$. Test at the $5 \%$ level of significance whether it is reasonable to suppose that the model applies with $p = \frac { 1 } { 2 }$.\\
(ii) The model is refined by estimating $p$ from the data. Find the mean of the observed data and hence an estimate of $p$.\\
(iii) Using the estimated value of $p$, the value of the test statistic $X ^ { 2 }$ turns out to be 2.3857 . Is it reasonable to suppose, at the $5 \%$ level of significance, that this refined model applies?\\
(iv) Discuss the reasons for the different outcomes of the tests in parts (i) and (iii).
\hfill \mbox{\textit{OCR MEI S3 2010 Q1 [17]}}