Unbiased estimates calculation

4 The mean time to mark a certain set of examination papers is estimated by the examination board to be 12 minutes per paper. A random sample of 150 examination papers gave \(\Sigma x = 2130\) and \(\Sigma x ^ { 2 } = 37746\), where \(x\) is the time in minutes to mark an examination paper.

1. Calculate unbiased estimates of the population mean and variance.
2. Stating the null and alternative hypotheses, use a \(10 \%\) significance level to test whether the examination board's estimated time is consistent with the data.

4

1. A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows. $$\begin{array} { l l l l l l l l } 261 & 249 & 259 & 252 & 255 & 256 & 258 & 254 \end{array}$$ Find unbiased estimates of the population mean and variance.
2. The supplier claims that the mean mass of boxes of cereal is 253 g . A quality control officer suspects that the mean mass is actually more than 253 g . In order to test this claim, he weighs a random sample of 100 boxes of cereal and finds that the total mass is 25360 g .
   1. Given that the population standard deviation of the masses is 3.5 g , test at the \(5 \%\) significance level whether the population mean mass is more than 253 g .
      An employee says, 'This test is invalid because it uses the normal distribution, but we do not know whether the masses of the boxes are normally distributed.'
   2. Explain briefly whether this statement is true or not.

4 The height of sweet pea plants grown in a nursery is a random variable. A random sample of 50 plants is measured and is found to have a mean height 1.72 m and variance \(0.0967 \mathrm {~m} ^ { 2 }\).

1. Calculate an unbiased estimate for the population variance of the heights of sweet pea plants.
2. Hence test, at the \(10 \%\) significance level, whether the mean height of sweet pea plants grown by the nursery is 1.8 m , stating your hypotheses clearly.

1. The average time it takes for a new kettle to boil, when full of water, is 305 seconds. An old kettle will take longer, on average, to boil. Alun suspects that a particular kettle is an old kettle. He boils the full kettle on 9 occasions and the times taken, in seconds, are shown below.
   305
   295
   310
   310
   315
   307
   300
   311
   306

You may assume the times taken to boil the full kettle are normally distributed.

1. Calculate unbiased estimates for the mean and variance of the times taken to boil the full kettle.
2. Test, at the \(5 \%\) level of significance, whether there is evidence to suggest that this is an old kettle.
3. State a factor that Alun should control when carrying out this investigation.

1. Kaff coffee is sold in packets. A seller measures the masses of the contents of a random sample of 90 packets of Kaff coffee from her stock. The results are shown in the table below.

$$\text { (You may use } \sum \mathrm { fy } ^ { 2 } = 5644 \text { 171.75) }$$ A histogram is drawn and the class \(245 \leq w < 248\) is represented by a rectangle of width 1.2 cm and height 10 cm .

1. Calculate the width and the height of the rectangle representing the class \(255 \leq w < 260\).
2. Use linear interpolation to estimate the median mass of the contents of a packet of Kaff coffee to 1 decimal place.
3. Estimate the mean and the standard deviation of the mass of the contents of a packet of Kaff coffee to 1 decimal place. The seller claims that the mean mass of the contents of the packets is more than the stated mass. Given that the stated mass of the contents of a packet of Kaff coffee is 250 g and the actual standard deviation of the contents of a packet of Kaff coffee is 4 g ,
4. test, using a 5\% level of significance, whether or not the seller's claim is justified. State your hypotheses clearly.
   (You may assume that the mass of the contents of a packet is normally distributed.)
5. Using your answers to parts (b) and (c), comment on the assumption that the mass of the contents of a packet is normally distributed.
   (Total 14 marks)