Binomial parameters from given information

4

1. In Germany, towards the end of the nineteenth century, a study was undertaken into the distribution of the sexes in families of various sizes. The table shows some data about the numbers of girls in 500 families, each with 5 children. It is thought that the binomial distribution \(\mathrm { B } ( 5 , p )\) should model these data.

   Number of girls	Number of families
   0	32
   1	110
   2	154
   3	125
   4	63
   5	16

   1. Use this information to calculate an estimate for the mean number of girls per family of 5 children. Hence show that 0.45 can be taken as an estimate of \(p\).
   2. Investigate at a \(5 \%\) significance level whether the binomial model with \(p\) estimated as 0.45 fits the data. Comment on your findings and also on the extent to which the conditions for a binomial model are likely to be met.
2. A researcher wishes to select 50 families from the 500 in part (a) for further study. Suggest what sort of sample she might choose and describe how she should go about choosing it.

13 At a certain factory Christmas tree decorations are packed in boxes of 10 . The quality control manager collects a random sample of 100 boxes of decorations and records the number of decorations in each box which are damaged. His results are displayed in Fig. 13.1. \begin{table}[h] \end{table}

1. Calculate
   • the mean number of damaged decorations per box,
   • the standard deviation of the number of damaged decorations per box.
   It is believed that the number of damaged decorations in a box of 10, \(X\), may be modelled by a binomial distribution such that \(\mathrm { X } \sim \mathrm { B } ( \mathrm { n } , \mathrm { p } )\).
2. State suitable values for \(n\) and \(p\).
3. Use the binomial model to complete the copy of Fig. 13.2 in the Printed Answer Booklet, giving your answers correct to \(\mathbf { 1 }\) decimal place. \begin{table}[h]

   Number of damaged decorations	0	1	2	3	4	5 or more
   Observed number of boxes	19	35	28	13	5	0
   Expected number of boxes

   \captionsetup{labelformat=empty} \caption{Fig. 13.2}
   \end{table}
4. Explain whether the model is a good fit for these data.

4. A sweet shop produces different coloured sweets and sells them in bags. The proportion of green sweets produced is \(p\) Each bag is filled with a random sample of \(n\) sweets. The mean number of green sweets in a bag is 4.2 and the variance is 3.57

1. Find the value of \(n\) and the value of \(p\) The proportion of red sweets produced by the shop is 0.35
2. Find the probability that, in a random sample of 25 sweets, the number of red sweets exceeds the expected number of red sweets. The shop claims that \(10 \%\) of its customers buy more than two bags of sweets. A random sample of 40 customers is taken and 1 customer buys more than two bags of sweets.
3. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the proportion of customers who buy more than two bags of sweets is less than the shop's claim. State your hypotheses clearly.