CI from raw data list

A machine is filling bottles of milk. A random sample of 16 bottles was taken and the volume of milk in each bottle was measured and recorded. The volume of milk in a bottle is normally distributed and the unbiased estimate of the variance, \(s^2\), of the volume of milk in a bottle is 0.003

1. Find a 95\% confidence interval for the variance of the population of volumes of milk from which the sample was taken. [5] The machine should fill bottles so that the standard deviation of the volumes is equal to 0.07
2. Comment on this with reference to your 95\% confidence interval. [3]

A random sample of 15 strawberries is taken from a large field and the weight \(x\) grams of each strawberry is recorded. The results are summarised below. $$\sum x = 291 \quad \sum x^2 = 5968$$ Assume that the weights of strawberries are normally distributed. Calculate a 95\% confidence interval for

1. (i) the mean of the weights of the strawberries in the field, (ii) the variance of the weights of the strawberries in the field. [12]

Strawberries weighing more than 23g are considered to be less tasty.

1. Use appropriate confidence limits from part (a) to find the highest estimate of the proportion of strawberries that are considered to be less tasty. [4]

Two independent random samples \(X_1, X_2, ..., X_n\) and \(Y_1, Y_2, Y_3, Y_4\) were taken from different normal populations with a common standard deviation \(\sigma\). The following sample statistics were calculated. $$s_x = 14.67 \quad s_y = 12.07$$ Find the 99\% confidence interval for \(\sigma^2\) based on these two samples. [5]

A newspaper runs a daily Sudoku. A random sample of 10 people took the following times, in minutes, to complete the Sudoku. 5.0 \quad 4.5 \quad 4.7 \quad 5.3 \quad 5.2 \quad 4.1 \quad 5.3 \quad 4.8 \quad 5.5 \quad 4.6 Given that the times to complete the Sudoku follow a normal distribution,

1. calculate a 95\% confidence interval for
   1. the mean,
   2. the variance,
   of the times taken by people to complete the Sudoku. [13] The newspaper requires the average time needed to complete the Sudoku to be 5 minutes with a standard deviation of 0.7 minutes.
2. Comment on whether or not the Sudoku meets this requirement. Give a reason for your answer. [3]

A coffee shop produces biscuits to sell. The masses, in grams, of the biscuits follow a normal distribution with mean \(\mu\). Eight biscuits are chosen at random and their masses, in grams, are recorded. The results are given below. 32.1 \quad 29.9 \quad 31.0 \quad 31.1 \quad 32.5 \quad 30.8 \quad 30.7 \quad 31.5

1. Calculate a 95\% confidence interval for \(\mu\) based on this sample. [7]
2. Explain the relevance or otherwise of the Central Limit Theorem in your calculations. [1]

A factory manufactures a certain type of string. In order to ensure the quality of the product, a random sample of 10 pieces of string is taken every morning and the breaking strength of each piece, in Newtons, is measured. One morning, the results are as follows. $$68.1 \quad 70.4 \quad 68.6 \quad 67.7 \quad 71.3 \quad 67.6 \quad 68.9 \quad 70.2 \quad 68.4 \quad 69.8$$ You may assume that this is a random sample from a normal distribution with unknown mean \(\mu\) and unknown variance \(\sigma^2\).

1. Determine a 95% confidence interval for \(\mu\). [9]
2. The factory manager is given these results and he asks 'Can I assume that the confidence interval that you have given me contains \(\mu\) with probability 0.95?' Explain why the answer to this question is no and give a correct interpretation. [2]

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.