Percentages given, table construction required

1. A Head of Department at a large university believes that gender is independent of the grade obtained by students on a Business Foundation course. A random sample was taken of 200 male students and 160 female students who had studied the course.

The results are summarised below. Stating your hypotheses clearly, test the Head of Department's belief using a \(5 \%\) level of significance. Show your working clearly.

5. A random sample of 500 adults completed a questionnaire on how often they took part in some form of exercise. They gave a response of 'never', 'sometimes' or 'regularly'. Of those asked, \(52 \%\) were females of whom \(10 \%\) never exercised and \(35 \%\) exercised regularly. Of the males, \(12.5 \%\) never exercised and \(55 \%\) sometimes exercised. Test, at the \(5 \%\) level of significance, whether or not there is any association between gender and the amount of exercise. State your hypotheses clearly.

2 It is claimed that the way in which students voted at a particular general election was independent of their gender. In order to investigate this claim, 480 male and 540 female students who voted at this general election were surveyed. These students may be regarded as a random sample. The percentages of males and females who voted for the different parties are recorded in the table.

1. Complete the contingency table below.
2. Hence determine, at the \(1 \%\) level of significance, whether the way in which students voted at this general election was independent of their gender.

   	Conservative	Labour	Liberal Democrat	Other parties	Total
   Male					480
   Female					540
   Total					1020

4 It is claimed that the area within which a school is situated affects the age profile of the staff employed at that school. In order to investigate this claim, the age profiles of staff employed at two schools with similar academic achievements are compared. Academia High School, situated in a rural community, employs 120 staff whilst Best Manor Grammar School, situated in an inner-city community, employs 80 staff. The percentage of staff within each age group, for each school, is given in the table.

1. 1. Form the data into a contingency table suitable for analysis using a \(\chi ^ { 2 }\) distribution.
      (2 marks)
   2. Use a \(\chi ^ { 2 }\) test, at the \(1 \%\) level of significance, to determine whether there is an association between the age profile of the staff employed and the area within which the school is situated.
2. Interpret your result in part (a)(ii) as it relates to the 22-34 age group.

9 A company owns three shops, A, B and C, which are based in different towns. Each shop gives a questionnaire to 250 of their customers, and every customer completes the questionnaire. One of the questions asks whether the customer rates the shop as good, satisfactory or poor. For shop A, 26\% of customers rate the shop as good and 38\% of customers rate the shop as poor. For shop B, 32\% of customers rate the shop as good and 40\% of customers rate the shop as satisfactory. Altogether, there are 210 good ratings and 261 satisfactory ratings. 9

1. Complete the following table with the observed frequencies.

   \multirow{2}{*}{}		Rating
   		Good	Satisfactory	Poor
   \multirow{3}{*}{Shop}	A
   	B
   	C

   9
2. Carry out a test for association between shop and rating, using the 1\% level of significance.