There are a large number of students at a particular college. The heights, in metres, of a random sample of 8 students are as follows.
\[\begin{array}{llllllll}
1.75 & 1.72 & 1.62 & 1.70 & 1.82 & 1.75 & 1.68 & 1.84
\end{array}\]

You may assume that heights of students are normally distributed.
(i) Test, at the \(5 \%\) significance level, whether the population mean height of students at this college is greater than 1.70 metres.

(ii) Find a 95\% confidence interval for the population mean height of students at this college.

A farmer grows large amounts of a certain crop. On average, the yield per plant has been 0.75 kg . The farmer has improved the soil in which the crop grows, and she claims that the yield per plant has increased. A random sample of 10 plants grown in the improved soil is chosen. The yields, \(x \mathrm{~kg}\), are summarised as follows.
\[\Sigma x=7.85 \quad \Sigma x^{2}=6.19\]
(i) Test at the 5\% significance level whether the farmer's claim is justified, assuming a normal distribution.
(ii) Find a 95\% confidence interval for the population mean yield for plants grown in the new soil.

A farmer grows a particular type of fruit tree. On average, the mass of fruit produced per tree has been 6.2 kg . He has developed a new kind of soil and claims that the mean mass of fruit produced per tree when growing in this new soil has increased. A random sample of 10 trees grown in the new soil is chosen. The masses, \(x \mathrm{~kg}\), of fruit produced are summarised as follows.
\[\Sigma x=72.0 \quad \Sigma x^{2}=542.0\]

Test at the 5\% significance level whether the farmer's claim is justified, assuming a normal distribution.

A large number of athletes are taking part in a competition. The masses, in kg , of a random sample of 7 athletes are as follows.
\[\begin{array}{lllllll}
98.1 & 105.0 & 92.2 & 89.8 & 99.9 & 95.4 & 101.2
\end{array}\]

Assuming that masses are normally distributed, test, at the \(10 \%\) significance level, whether the mean mass of athletes in this competition is equal to 94 kg .

A doctor is investigating the concentration of blood glucose in patients at risk of developing type 2 diabetes, where blood glucose is measured in appropriate units. The doctor claims that a particular intervention reduces the concentration by more than \(k\) units on average. A group of 8 at risk patients is selected at random and each patient follows the intervention for six months. The blood glucose concentrations before and after the intervention are given in the following table.

(b) State an assumption necessary for the test in part (a) to be valid.

Maya is an athlete who competes in 1500 -metre races. Last summer her practice run times had mean 4.22 minutes. Over the winter she has done some intense training to try to improve her times. A random sample of 10 of her practice run times, \(x\) minutes, this summer are summarised as follows.
\(\sum x=42.05 \quad \sum x^{2}=176.83\)

Maya's new practice run times are normally distributed. She believes that on average her times have improved as a result of her training.

Test, at the \(5 \%\) significance level, whether Maya's belief is supported by the data.

A random sample of 7 observations of a variable \(X\) are as follows.

The population mean of \(X\) is \(\mu\).
(a) Test, at the \(10 \%\) significance level, the null hypothesis \(\mu=8.22\) against the alternative hypothesis \(\mu\lt 8.22\).
(b) State an assumption necessary for the test in part (a) to be valid.

Farmer \(A\) grows apples of a certain variety. Each tree produces 14.8 kg of apples, on average, per year. Farmer \(B\) grows apples of the same variety and claims that his apple trees produce a higher mass of apples per year than Farmer \(A\) 's trees. The masses of apples from Farmer \(B\) 's trees may be assumed to be normally distributed.

A random sample of 10 trees from Farmer \(B\) is chosen. The masses, \(x \mathrm{~kg}\), of apples produced in a year are summarised as follows.
\(\sum x=152.0 \quad \sum x^{2}=2313.0\)

Test, at the \(5 \%\) significance level, whether Farmer \(B\) 's claim is justified.

Manet has developed a new training course to help athletes improve their time taken to run 800 m . Manet claims that his course will decrease an athlete's time by more than 2 s on average. For a random sample of 10 athletes the times taken, in seconds, before and after the course are given in the following table.

Use a \(t\)-test, at the \(5 \%\) significance level, to test whether Manet's claim is justified, stating any assumption that you make.