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.
\(\begin{array} { l l l l l l l l } 32 \cdot 1 & 29 \cdot 9 & 31 \cdot 0 & 31 \cdot 1 & 32 \cdot 5 & 30 \cdot 8 & 30 \cdot 7 & 31 \cdot 5 \end{array}\)
Calculate a 95\% confidence interval for \(\mu\) based on this sample.
Explain the relevance or otherwise of the Central Limit Theorem in your calculations.
The continuous random variable \(X\) is uniformly distributed over the interval \(( \theta - 1 , \theta + 5 )\), where \(\theta\) is an unknown constant.
Find the mean and the variance of \(X\).
Let \(\bar { X }\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\bar { X }\), an unbiased estimator for \(\theta\) and determine its standard error.
The rules for the weight of a cricket ball state:
"The ball, when new, shall weigh not less than \(\mathbf { 1 5 5 . 9 ~ g }\), nor more than \(\mathbf { 1 6 3 ~ g }\)."
A company produces cricket balls whose weights are normally distributed. It wants \(99 \%\) of the balls it produces to be an acceptable weight.
What is the largest acceptable standard deviation?
The weights of the cricket balls are in fact normally distributed with mean 159.5 grams and standard deviation 1.2 grams. The company also produces tennis balls. The weights of the tennis balls are normally distributed with mean 58.5 grams and standard deviation 1.3 grams.
Find the probability that the weight of a randomly chosen cricket ball is more than three times the weight of a randomly chosen tennis ball.