10 A scientist is researching dietary fat intake and cholesterol level. A random sample of 60 people is selected and their dietary fat intakes and cholesterol levels are measured. Dietary fat intakes are classified as low, medium and high, and cholesterol levels are classified as normal and high.
The scientist decides to carry out a chi-squared test to investigate whether there is any association between dietary fat intake and cholesterol level. Tables \(\mathbf { 1 0 . 1 }\) and \(\mathbf { 1 0 . 2 }\) show the data and some of the expected frequencies for the test.
\begin{table}[h]
| \multirow{2}{*}{} | Dietary fat intake | |
| | Low | Medium | High | Total |
| \multirow{2}{*}{Cholesterol level} | Normal | 9 | 18 | 5 | 32 |
| High | 3 | 13 | 12 | 28 |
| Total | 12 | 31 | 17 | 60 |
\captionsetup{labelformat=empty}
\caption{Table 10.1}
\end{table}
\begin{table}[h]
| Expected frequency | Dietary fat intake |
| \cline { 3 - 5 } | Low | Medium | High | |
| \multirow{2}{*}{} | Normal | | | 9.0667 |
| \cline { 2 - 5 } | High | | | 7.9333 |
\captionsetup{labelformat=empty}
\caption{Table 10.2}
\end{table}
- Complete the table of expected frequencies in the Printed Answer Booklet.
- Determine the contribution to the chi-squared test statistic for people with normal cholesterol level and high dietary fat intake, giving your answer to \(\mathbf { 4 }\) decimal places.
The contributions to the chi-squared test statistic for the remaining categories are shown in Table 10.3.
\begin{table}[h]
| Dietary fat intake | | \cline { 2 - 5 } | Low | Medium | High | | | \multirow{2}{*}{} | Normal | 1.0563 | 0.1301 | | | \cline { 2 - 5 } | High | 1.2071 | 0.1487 | 2.0846 |
\captionsetup{labelformat=empty}
\caption{Table 10.3}
\end{table}- In this question you must show detailed reasoning.
Carry out the test at the 5\% significance level.
- For each level of dietary fat intake, give a brief interpretation of what the data suggest about the level of cholesterol.
11 A particular dietary supplement, when taken for a period of 1 month, is claimed to increase lean body mass of adults by an average of 1 kg . A researcher believes that this claim overestimates the increase. She selects a random sample of 10 adults who then each take the supplement for a month. The increases in lean body masses in kg are as follows.
$$\begin{array} { l l l l l l l l l l }
- 0.84 & - 0.76 & - 0.16 & 0.43 & 1.31 & 1.32 & 1.47 & 1.64 & 1.93 & 2.14
\end{array}$$
A Normal probability plot and the \(p\)-value of the Kolmogorov-Smirnov test for these data are shown below.
\includegraphics[max width=\textwidth, alt={}, center]{77eabbd6-a058-457f-9601-d66f3c2db005-09_575_1485_689_242}
- The researcher decides to carry out a hypothesis test in order to investigate the claim.
Comment on the type of hypothesis test that should be used. You should refer to
12 The continuous random variable \(X\) has cumulative distribution function given by
$$F ( x ) = \begin{cases} 0 & x < 0 \\ k \left( a x - 0.5 x ^ { 2 } \right) & 0 \leqslant x \leqslant a \\ 1 & x > a \end{cases}$$
where \(a\) and \(k\) are positive constants.
- Determine the median of \(X\) in terms of \(a\).
- Given that \(a = 10\), determine the probability that \(X\) is within one standard deviation of its mean.
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