Larger contingency table (4+ categories)

2 A driving school employs four instructors to prepare people for their driving test. The allocation of people to instructors is random. For each of the instructors, the following table gives the number of people who passed and the number who failed their driving test last year. Test at the 10\% significance level whether success in the driving test is independent of the instructor.

2 A scientist is investigating the size of shells at various beach locations. She selects four beach locations and takes a random sample of shells from each of these beaches. She classifies each shell as large or small. Her results are summarised in the following table. Test, at the 10\% significance level, whether the size of shell is independent of the beach location.

2 A town council has published its plans for redeveloping the town centre and residents are being asked whether they approve or disapprove. A random sample of 250 responses has been selected from residents in the four main streets in the town: North, East, South and West Streets. The results are shown in the table. Test, at the \(5 \%\) significance level, whether the opinions of the residents are independent of the streets on which they live.

4

1. A researcher believes that there may be some association between a student's sex and choice of certain subjects at A-level. A random sample of 250 A -level students is selected. The table below shows, for each sex, how many study either or both of the two subjects, Mathematics and English.

   	Mathematics only	English only	Both	Neither	Row totals
   Male	38	19	6	32	95
   Female	42	55	9	49	155
   Column totals	80	74	15	81	250

   Carry out a test at the \(5 \%\) significance level to examine whether there is any association between a student's sex and choice of subjects. State carefully your null and alternative hypotheses. Your working should include a table showing the contributions of each cell to the test statistic. [12]
2. Over a long period it has been determined that the mean score of students in a particular English module is 67.4 and the standard deviation is 8.9. A new teaching method is introduced with the aim of improving the results. A random sample of 12 students taught by the new method is selected. Their mean score is found to be 68.3. Carry out a test at the \(10 \%\) level to investigate whether the new method appears to have been successful. State carefully your null and alternative hypotheses. You should assume that the scores are Normally distributed and that the standard deviation is unchanged.

4 A genetics researcher is investigating whether there is any association between natural hair colour and natural eye colour. A random sample of 800 adults is selected. Each adult can categorise their natural hair colour as blonde, brown, black or red and their natural eye colour as brown, blue or green.

1. Explain the benefit of using a random sample in this investigation. The data collected from the sample are summarised in Table 4.1. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.1}

   \multirow{2}{*}{Observed frequency}		Hair Colour
   		Blonde	Brown	Black	Red	Total
   \multirow{3}{*}{Eye Colour}	Brown	47	153	196	36	432
   	Blue	61	78	115	26	280
   	Green	19	22	31	16	88
   	Total	127	253	342	78	800

   \end{table} The researcher decides to carry out a chi-squared test.
2. Determine the expected frequencies for each eye colour in the blonde hair category. You are given that the test statistic is 28.62 to 2 decimal places.
3. Carry out the chi-squared test at the 10\% significance level. Table 4.2 shows the chi-squared contributions for some of the categories. The contributions for the categories relating to green eye colour have been deliberately omitted. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Table 4.2}

   Hair Colour \cline { 2 - 6 } Blonde Brown Black Red \multirow{3}{*}{ Eye Colour } Brown 6.791 1.964 0.694 0.889 \cline { 2 - 6 } Blue 6.162 1.257 0.185 0.062 \cline { 2 - 6 } Green \end{table} Calculate the chi-squared contribution for the green eye and blonde hair category. With reference to the values in Table 4.2, discuss what the data suggest about brown eye colour and blue eye colour for people with blonde hair. A different researcher, carrying out the same investigation, independently takes a different random sample of size 800 and performs the same hypothesis test, but at the 1\% significance level, reaching the same conclusion as the original test. By comparing only the significance level of the two tests, specify which test, the one at the 10\% significance level or the one at the 1\% significance level, provides stronger evidence for the conclusion. Justify your answer. Edexcel FS1 AS 2020 June Q2 15 marks Standard +0.3 In an experiment, James flips a coin 3 times and records the number of heads. He carries out the experiment 100 times with his left hand and 100 times with his right hand. \multirow{2}{*}{} Number of heads 0 1 2 3 Left hand 7 29 42 22 Right hand 13 35 36 16 Test, at the \(5 \%\) level of significance, whether or not there is an association between the hand he flips the coin with and the number of heads. You should state your hypotheses, the degrees of freedom and the critical value used for this test. Assuming the coin is unbiased, write down the distribution of the number of heads in 3 flips. Carry out a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether or not the distribution you wrote down in part (b) is a suitable model for the number of heads obtained in the 200 trials of James' experiment. You should state your hypotheses, the degrees of freedom and the critical value used for this test. AQA S2 2009 January Q1 11 marks Standard +0.3 1 Fortune High School gave its students a wider choice of subjects to study. The table shows the number of students, of each gender, who chose to study each of the additional subjects during the school year 2007/08. \cline { 2 - 5 } \multicolumn{1}{c|}{} Bulgarian Climate Change Finance Polish Male 7 31 25 40 Female 2 24 22 19 Assuming that these data form a random sample, use a \(\chi ^ { 2 }\) test, at the \(10 \%\) level of significance, to test whether the choice of these subjects is independent of gender. (11 marks) AQA Further AS Paper 2 Statistics 2020 June Q2 1 marks Moderate -0.8 A \(\chi^2\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test. Number of times late 0 1 2 3 4 A 8.12 14 15.12 14 4.76 Class B 8.99 15.5 16.74 15.5 5.27 C 11.89 20.5 22.14 20.5 6.97 Find a possible value for the degrees of freedom for the test. Circle your answer. [1 mark] 6 \quad 8 \quad 12 \quad 15