1. A psychologist claims that the mean age at which female children start walking is 11.4 months. If 20 randomly selected female children are found to have started walking at a mean age of 11.5 months with standard deviation of 2 months, would you agree with the psychologist's claim? Assume that the sample came from a normal population.
2. Let U1 and U2 be independent random variables. Suppose that U1 is χ2 with ν1 degrees of freedom while U = U1 + U2 is chi-square with ν degrees of freedom, where ν > ν1. Then prove that U2 is chi-square random variable with ν - ν1 degrees of freedom.