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Exam-Style Question on SignificanceA mathematics exam-style question with a worked solution that can be revealed gradually |
Question id: 26. This question is similar to one that appeared on an IB Studies paper in 2012. The use of a calculator is allowed.
The staff of a shop kept records of who bought smart phones during the month of February one year. They looked at the numbers of phones bought by gender and the size of the screens. This information is shown in the table below; S represents the size of the screen in centimetres.
| S ≤ 12 | 12 < S ≤ 16 | 16 < S ≤ 20 | S > 20 | Total | |
|---|---|---|---|---|---|
| Female | 78 | 113 | 53 | 28 | 272 |
| Male | 33 | 78 | 153 | 68 | 332 |
| Total | 111 | 191 | 206 | 96 | 604 |
The shop manager wants to use this information to predict the probability of selling these sizes of phone screens for the following month.
(a) Use the table to find the probability that a phone will be bought by a female.
(b) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought.
(c) Find the probability that a phone with a screen size of 12cm < S ≤ 16cm will be bought by a female.
(d) Find the probability that a phone with a screen size greater than 20cm will be bought given that it is bought by a male.
The manager wants to determine whether the screen size is independent of gender so a chi-squared test is performed at the 1% significance level.
(e) Write down the null hypothesis.
(f) Find the expected frequency for females who bought a screen size of 12cm < S ≤ 16cm to the nearest integer.
(g) Write down the number of degrees of freedom.
(h) Write down the \(\chi ^2\) calculated value.
(i) Determine if the null hypothesis should be accepted. Give a reason for your answer.
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