2 The head circumference of 3-year-old boys is known to be Normally distributed with mean 49.7 cm and standard deviation 1.6 cm .
- Find the probability that the head circumference of a randomly selected 3 -year-old boy will be
(A) over 51.5 cm ,
(B) between 48.0 and 51.5 cm . - Four 3-year-old boys are selected at random. Find the probability that exactly one of them has head circumference between 48.0 and 51.5 cm .
- The head circumference of 3-year-old girls is known to be Normally distributed with mean \(\mu\) and standard deviation \(\sigma\). Given that \(60 \%\) of 3-year-old girls have head circumference below 49.0 cm and \(30 \%\) have head circumference below 47.5 cm , find the values of \(\mu\) and \(\sigma\).
A nutritionist claims that boys who have been fed on a special organic diet will have a larger mean head circumference than other boys. A random sample of ten 3 -year-old boys who have been fed on this organic diet is selected. It is found that their mean head circumference is 50.45 cm .
- Using the null and alternative hypotheses \(\mathrm { H } _ { 0 } : \mu = 49.7 \mathrm {~cm} , \mathrm { H } _ { 1 } : \mu > 49.7 \mathrm {~cm}\), carry out a test at the \(10 \%\) significance level to examine the nutritionist's claim. Explain the meaning of \(\mu\) in these hypotheses. You may assume that the standard deviation of the head circumference of organically fed 3 -year-old boys is 1.6 cm .