5.05c Hypothesis test: normal distribution for population mean

A new species of animal has been found on an uninhabited island. A zoologist wishes to investigate whether or not there is a difference in the mean weights of males and females of the species. She traps some of the animals and weighs them with the following results. \begin{align} \text{Males (kg)} &\quad 5.3, 4.6, 5.2, 4.5, 4.3, 5.5, 5.0, 4.8
\text{Females (kg)} &\quad 4.9, 5.0, 4.1, 4.6, 4.3, 5.3, 4.2, 4.5, 4.8, 4.9 \end{align} You may assume that these are random samples from normal populations with a common standard deviation of 0.5 kg.

1. State suitable hypotheses for this investigation. [1]
2. Determine the \(p\)-value of these results and state your conclusion in context. [9]

A shopkeeper sells chocolate bars which are described by the manufacturer as having an average mass of 45 grams. The shopkeeper claims that the mass of the chocolate bars, \(X\) grams, is getting smaller on average. A random sample of 6 chocolate bars is taken and their masses in grams are measured. The results are $$\sum x = 246 \quad \text{and} \quad \sum x^2 = 10198$$ Investigate the shopkeeper's claim using the 5\% level of significance. State any assumptions that you make. [9 marks]

The greatest weight \(W\) N that can be supported by a shelving bracket of traditional design is a normally distributed random variable with mean 500 and standard deviation 80. A sample of 40 shelving brackets of a new design are tested and it is found that the mean of the greatest weights that the brackets in the sample can support is 473.0 N.

1. Test at the 1% significance level whether the mean of the greatest weight that a bracket of the new design can support is less than the mean of the greatest weight that a bracket of the traditional design can support. [7]
2. State an assumption needed in carrying out the test in part (a). [1]
3. Explain whether it is necessary to use the central limit theorem in carrying out the test. [1]

Sweet pea plants grown using a standard plant food have a mean height of 1.6 m. A new plant food is used for a random sample of 49 randomly chosen plants and the heights, \(x\) metres, of this sample can be summarised by the following. $$n = 49$$ $$\sum x = 74.48$$ $$\sum x^2 = 120.8896$$ Test, at the 5\% significance level, whether, when the new plant food is used, the mean height of sweet pea plants is less than 1.6 m. [9]

11 Answer only one of the following two alternatives.
EITHER
A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.
OR
A factory produces bottles of spring water. The manager decides to assess the performance of the two machines that are used to fill the bottles with water. He selects a random sample of 60 bottles filled by the first machine \(X\) and a random sample of 80 bottles filled by the second machine \(Y\). The volumes of water, \(x\) and \(y\), measured in appropriate units, are summarised as follows. $$\Sigma x = 58.2 \quad \Sigma x ^ { 2 } = 85.8 \quad \Sigma y = 97.6 \quad \Sigma y ^ { 2 } = 188.6$$ A test at the \(\alpha \%\) significance level shows that the mean volume of water in bottles filled by machine \(X\) is less than the mean volume of water in bottles filled by machine \(Y\). Find the set of possible values of \(\alpha\).

10 The lengths of a random sample of eight fish of a certain species are measured, in cm, as follows. $$\begin{array} { l l l l l l l l } 17.3 & 15.8 & 18.2 & 15.6 & 16.0 & 18.8 & 15.3 & 15.0 \end{array}$$ Assuming that lengths are normally distributed,

1. test, at the \(10 \%\) significance level, whether the population mean length of fish of this species is greater than 15.8 cm ,
2. calculate a \(95 \%\) confidence interval for the population mean length of fish of this species.