Further Statistics 1: Chi squared testsEdexcel A-Level Further Maths: Topic test

1. 1
   A spinner has five equal sectors numbered $1$ to $5$. It is spun $100$ times and the frequencies of the scores $1,2,3,4,5$ are $25,18,22,15,20$ respectively. A chi-squared test is carried out at the $5\%$ significance level to test whether the scores follow a discrete uniform distribution.

   (a)

   What is the expected frequency of each score under the null hypothesis?

   [1 mark]

   • A$5$
   • B$20$
   • C$25$
   • D$22$

   (b)

   How many degrees of freedom does the test have?

   [1 mark]

   • A$4$
   • B$5$
   • C$3$
   • D$99$

   (c)

   Calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A plant breeder crosses plants in sets of $3$ seedlings, and each seedling independently has white flowers with probability $0.25$. In $120$ sets, the numbers of seedlings with white flowers were: $0$ white in $46$ sets, $1$ white in $55$ sets, $2$ white in $17$ sets, and $3$ white in $2$ sets. A chi-squared goodness of fit test at the $5\%$ level is used to see whether $\mathrm{B}(3,0.25)$ is a suitable model.

   (a)

   What is the expected number of sets with exactly $1$ white seedling?

   [1 mark]

   • A$30$
   • B$55$
   • C$50.625$
   • D$0.4219$

   (b)

   Which classes must be combined before the test statistic is calculated?

   [1 mark]

   • ANone of the classes
   • BThe classes with $0$ and $1$ white seedlings
   • CThe classes with $1$ and $2$ white seedlings
   • DThe classes with $2$ and $3$ white seedlings

   (c)

   Combine classes where necessary and calculate the value of the test statistic $\sum\frac{(O_i-E_i)^2}{E_i}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bus stop is observed during $100$ ten-minute intervals, and the number of buses arriving in each interval is recorded: $0$ buses in $20$ intervals, $1$ bus in $35$, $2$ buses in $27$, $3$ buses in $12$, $4$ buses in $5$ and $5$ buses in $1$ interval. A researcher tests, at the $5\%$ level, whether a Poisson distribution is a suitable model.

   (a)

   Show that the mean number of buses per interval is $1.5$, and find the expected number of intervals with exactly $3$ buses for a Poisson distribution with this mean.

   [3 marks]

   (b)

   The expected frequencies for $0,1,2,3$ buses are $22.31$, $33.47$, $25.10$ and $12.55$, and the final two classes are combined as '$4$ or more' with observed frequency $6$ and expected frequency $6.56$. Carry out the test, stating the degrees of freedom and your conclusion.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A gym asks $150$ members for their membership type and preferred session time. Of the $90$ standard members, $24$ prefer mornings, $31$ afternoons and $35$ evenings. Of the $60$ premium members, $28$ prefer mornings, $14$ afternoons and $18$ evenings. The gym uses a chi-squared test at the $5\%$ level to test whether membership type and preferred session time are associated.

   (a)

   (i) State the null and alternative hypotheses. (ii) Show that the expected frequency for premium members who prefer evenings is $21.2$. (iii) State the number of degrees of freedom. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   The expected frequencies for standard members are $31.2$, $27.0$ and $31.8$ (morning, afternoon, evening) and for premium members $20.8$, $18.0$ and $21.2$. (i) Calculate the test statistic. (ii) The critical value at the $5\%$ level is $5.991$. State the conclusion in context. Marks: (i) 3, (ii) 3.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   In each of $100$ practice sessions a basketball player takes $5$ free throws, and the number of successful throws is recorded: $0$ successes in $6$ sessions, $1$ in $22$, $2$ in $34$, $3$ in $26$, $4$ in $10$ and $5$ in $2$ sessions. A coach tests, at the $5\%$ level, whether the number of successes in a session follows a binomial distribution $\mathrm{B}(5,p)$ with $p$ estimated from the data.

   (a)

   What is the estimate of $p$?

   [1 mark]

   • A$2.18$
   • B$0.436$
   • C$0.218$
   • D$0.564$

   (b)

   After any necessary combining of classes, there are $5$ classes. How many degrees of freedom does the test have?

   [1 mark]

   • A$3$
   • B$4$
   • C$5$
   • D$6$

   (c)

   Show that the class '$5$ successes' cannot be used on its own in the test, and state what should be done.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A veterinary clinic records whether $80$ dogs and $60$ cats are overweight. Of the dogs, $28$ are overweight, and of the cats, $12$ are overweight. The clinic uses a chi-squared test at the $5\%$ level to test whether being overweight is associated with the type of animal.

   (a)

   What is the expected number of overweight dogs if there is no association?

   [1 mark]

   • A$28$
   • B$20$
   • C$40$
   • D$22.9$

   (b)

   How many degrees of freedom does the test have?

   [1 mark]

   • A$4$
   • B$2$
   • C$1$
   • D$140$

   (c)

   The test statistic is $3.78$ and the critical value at the $5\%$ level is $3.841$. State the conclusion of the test in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A card is drawn at random from a standard pack, noted and replaced, until a heart is drawn. Each of $120$ players does this, and the number of draws needed is recorded: $1$ draw for $36$ players, $2$ draws for $25$, $3$ for $14$, $4$ for $15$, $5$ for $10$, and $6$ or more for $20$ players. A chi-squared test at the $5\%$ level is used to see whether a geometric distribution with $p=0.25$ is a suitable model.

   (a)

   Find the expected number of players who need exactly $3$ draws, and the expected number who need $6$ or more draws.

   [3 marks]

   (b)

   The expected frequencies for $1,2,3,4,5$ and '$6$ or more' draws are $30$, $22.5$, $16.88$, $12.66$, $9.49$ and $28.48$. Carry out the test, stating the degrees of freedom and your conclusion in context.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A weather station records the number of lightning strikes within $5$ km in each of $80$ hours: $0$ strikes in $4$ hours, $1$ strike in $21$, $2$ in $19$, $3$ in $16$, $4$ in $10$, and $5$ or more in $10$ hours. A meteorologist claims that the number of strikes per hour follows a Poisson distribution with mean $2.5$, and a chi-squared test is carried out at the $5\%$ level.

   (a)

   (i) Find the expected number of hours with exactly $2$ strikes. (ii) Find the expected number of hours with $5$ or more strikes. (iii) State, with a reason, whether any classes need to be combined, given that the expected frequencies for $0,1,3$ and $4$ strikes are $6.57$, $16.42$, $17.10$ and $10.69$. Marks: (i) 2, (ii) 2, (iii) 2.

   [6 marks]

   (b)

   The expected frequencies for $0,1,2,3,4$ and '$5$ or more' strikes are $6.57$, $16.42$, $20.52$, $17.10$, $10.69$ and $8.71$. (i) Calculate the test statistic. (ii) State the degrees of freedom, and the conclusion of the test in context, given that the critical value is $11.07$. (iii) State how the degrees of freedom would change if the mean had been estimated from the data. Marks: (i) 3, (ii) 2, (iii) 1.

   [6 marks]

   Total for question 8: 12 marks

End of questions