5.05b Unbiased estimates: of population mean and variance

A medical student is investigating whether there is a difference in a person's blood pressure when sitting down and after standing up. She takes a random sample of 12 people and measures their blood pressure, in mmHg, when sitting down and after standing up. The results are shown below.

Person	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)	\(K\)	\(L\)
Sitting down	135	146	138	146	141	158	136	135	146	161	119	151
Standing up	131	147	132	140	138	160	127	136	142	154	130	144

The student decides to carry out a paired \(t\)-test to investigate whether, on average, the blood pressure of a person when sitting down is more than their blood pressure after standing up.

1. State clearly the hypotheses that should be used and any necessary assumption that needs to be made. [2]
2. Carry out the test at the 1\% level of significance. [7]

A biologist investigating the shell size of turtles takes random samples of adult female and adult male turtles and records the length, \(x\) cm, of the shell. The results are summarised below.

	Number in sample	Sample mean \(\bar{x}\)	\(\sum x^2\)
Female	6	19.6	2308.01
Male	12	13.7	2262.57

You may assume that the samples come from independent normal distributions with the same variance. The biologist claims that the mean shell length of adult female turtles is 5 cm longer than the mean shell length of adult male turtles.

1. Test the biologist's claim at the 5\% level of significance. [10]
2. Given that the true values for the variance of the population of adult male turtles and adult female turtles are both 0.9 cm\(^2\),
   1. show that when samples of size 6 and 12 are used with a 5\% level of significance, the biologist's claim will be accepted if \(4.07 < \bar{X}_F - \bar{X}_M < 5.93\) where \(\bar{X}_F\) and \(\bar{X}_M\) are the mean shell lengths of females and males respectively.
   2. Hence find the probability of a type II error for this test if in fact the true mean shell length of adult female turtles is 6 cm more than the mean shell length of adult male turtles. [6]

Boxes of chocolates manufactured by Philippe have a mean weight of \(\mu\) grams and a standard deviation of \(\sigma\) grams. A random sample of 25 of these boxes are weighed. Using this sample, the unbiased estimate of \(\mu\) is 455 and the unbiased estimate of \(\sigma^2\) is 55.

1. Test, at the 5\% level of significance, whether or not \(\sigma\) is greater than 6. State your hypotheses clearly. [6]
2. Test, at the 5\% level of significance, whether or not \(\mu\) is more than 450. [6]
3. State an assumption you have made in order to carry out the above tests. [1]

There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day. Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time. On a randomly chosen day, he records a total of 16 births from the two hospitals.

1. Investigate Stuart's claim, using a suitable test at the 5% level of significance. [6 marks]
2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. [3 marks]

The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.

1. Find the mean and the variance of \(X\). [2]
2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]

The random variable \(X\) has probability density function $$f(x) = 1 + \frac{3\lambda x}{2} \quad \text{for } -\frac{1}{2} \leqslant x \leqslant \frac{1}{2},$$ $$f(x) = 0 \quad \text{otherwise,}$$ where \(\lambda\) is an unknown parameter such that \(-1 \leqslant \lambda \leqslant 1\).

1. 1. Find E\((X)\) in terms of \(\lambda\).
   2. Show that \(\text{Var}(X) = \frac{16 - 3\lambda^2}{192}\). [6]
2. Show that P\((X > 0) = \frac{8 + 3\lambda}{16}\). [2]

In order to estimate \(\lambda\), \(n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\overline{X}\).

1. 1. Identify the distribution of \(Y\).
   2. Show that \(T_1\) is an unbiased estimator for \(\lambda\), where $$T_1 = \frac{16Y}{3n} - \frac{8}{3}.$$ [4]
2. 1. Show that \(\text{Var}(T_1) = \frac{64 - 9\lambda^2}{9n}\).
   2. Given that \(T_2\) is also an unbiased estimator for \(\lambda\), where $$T_2 = 8\overline{X},$$ find an expression for Var\((T_2)\) in terms of \(\lambda\) and \(n\).
   3. Hence, giving a reason, determine which is the better estimator, \(T_1\) or \(T_2\). [6]

The probability density function of the continuous random variable \(X\) is given by $$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$ $$f(x) = 0 \quad \text{otherwise.}$$ \(\overline{X}\) is the mean of a random sample of \(n\) observations of \(X\).

1. 1. Show that \(U = \frac{4\overline{X}}{3}\) is an unbiased estimator for \(\alpha\). [5]
   2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\)? [9]
2. \(\overline{X}_1\) and \(\overline{X}_2\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2\) is also an unbiased estimator for \(\alpha\).
   1. Show that \(\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}\). [4]
   2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator. [1]

The discrete random variable \(X\) has the following probability distribution, where \(\theta\) is an unknown parameter belonging to the interval \(\left(0, \frac{1}{3}\right)\).

1. Obtain an expression for \(E(X)\) in terms of \(\theta\) and show that $$\text{Var}(X) = 4\theta(3 - \theta).$$ [4] In order to estimate the value of \(\theta\), a random sample of \(n\) observations on \(X\) was obtained and \(\bar{X}\) denotes the sample mean.
2. 1. Show that $$V = \frac{\bar{X} - 3}{2}$$ is an unbiased estimator for \(\theta\).
   2. Find an expression for the variance of \(V\). [4]
3. Let \(Y\) denote the number of observations in the random sample that are equal to 1. Show that $$W = \frac{Y}{n}$$ is an unbiased estimator for \(\theta\) and find an expression for \(\text{Var}(W)\). [5]
4. Determine which of \(V\) and \(W\) is the better estimator, explaining your method clearly. [4]