5.06a Chi-squared: contingency tables

Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below.

	Black	Non-black
Men	69	71
Women	30	55

Test at the 5\% significance level whether Carl's belief is justified. [8]

A student, considering options for the future, collects data on education and salary. The table below shows the highest level of education attained and the salary bracket of a random sample of 664 people.

	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification	Total
Less than £20 000	18	32	20	28	10	108
£20 000 to £60 000	50	95	112	155	50	462
More than £60 000	3	22	29	35	5	94
Total	71	149	161	218	65	664

By conducting a chi-squared test for independence, the student investigates the relationship between the highest level of education attained and the salary earned.

1. State the null and alternative hypotheses. [1]
2. The table below shows the expected values. Calculate the value of \(k\). [2]

   Expected values	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
   Less than £20 000	\(k\)	24·23	26·19	35·46	10·57
   £20 000 to £60 000	49·40	103·67	112·02	151·68	45·23
   More than £60 000	10·05	21·09	22·79	30·86	9·20

3. The following computer output is obtained. Calculate the values of \(m\) and \(n\). [2]

   Chi Squared Contributions	Fewer than 5 GCSE	5 or more GCSE	3 A Levels	University degree	Post graduate qualification
   Less than £20 000	3·604530799	\(m\)	1·46165	1·5686	0·03098
   £20 000 to £60 000	0·007272735	0·72535	4E-06	0·07264	0·50396
   More than £60 000	4·946619863	0·03897	1·69081	0·55498	\(n\)

   X-squared = 19·61301, df = 8, p-value = 0·0119
4. 1. Without carrying out any further calculations, explain how X-squared = 19·61301 (the \(\chi^2\) test statistic) was calculated. [2]
   2. Comment on the values in the "Fewer than 5 GCSE" column of the table in part (c). [2]
5. The student says that the highest levels of education lead to the highest paying jobs. Comment on the accuracy of the student's statement. [1]

The Pew Research Center's Internet Project offers scholars access to raw data sets from their research. One of the Pew Research Center's projects was on teenagers and technology. A random sample of American families was selected to complete a questionnaire. For each of their children, between and including the ages of 13 and 15, parents of these families were asked: Do you know your child's password for any of [his/her] social media accounts? Responses to this question were received from 493 families. The table below provides a summary of their responses.

	Age (years)			\multirow{2}{*}{Total}
\cline{2-4} Parent know password	13	14	15
Yes	76	75	67	218
No	66	103	106	275
Total	142	178	173	493

1. A test for significance is to be undertaken to see whether there is an association between whether a parent knows any of their child's social media passwords and the age of the child.
   1. Clearly state the null and alternative hypotheses.
   2. Obtain the expected value that is missing from the table below, indicating clearly how it is calculated from the data values given in the table above.
   Expected values:

   	Age (years)
   \cline{2-4} Parent knows password	13	14	15
   Yes	62.79	78.71	76.50
   No		99.29	96.50

   1. Obtain the two chi-squared contributions that are missing from the table below.
   Chi-squared contributions:

   	Age (years)
   \cline{2-4} Parent knows password	13	14	15
   Yes		0.175	1.180
   No	2.203		0.935

   The following output was obtained from the statistical package that was used to undertake the analysis: Pearson chi-squared (2) = 7.409 \quad \(p\)-value = 0.0305
   1. Indicate how the degrees of freedom have been calculated for the chi-squared statistic.
   2. Interpret the output obtained from the statistical test in terms of the initial hypotheses. [10]
2. Comment on the nature of the association observed, based on the contributions to the test statistic calculated in (a). [2]

Carl believes that the proportions of men and women who own black cars are different. He obtained a random sample of people who each owned exactly one car. The results are summarised in the table below. Test at the 5\% significance level whether Carl's belief is justified. [8]