Practise selecting random samples, calculating unbiased estimates and sampling distributions and constructing or interpreting confidence intervals for population means.
The time, T minutes, for a certain daily bus journey is normally distributed. The bus company claims that the mean of T is 45. A passenger believes that the mean of T is actually greater than 45. She notes the times taken for this journey on a random sample of 60 days. The results are summarised below.
Calculate unbiased estimates of the population mean and variance.
Batteries of type A are known to have a mean life of 150 hours. It is required to test whether a new type of battery, type B, has a shorter mean life than type A batteries.
Give a reason for using a sample rather than the whole population in carrying out this test.
A random sample of 120 type B batteries are tested and it is found that their mean life is 147 hours, and an unbiased estimate of the population variance is 225 hours .
Calculate a 94% confidence interval for the population mean life of type B batteries.
Maroulla's calculator can generate random numbers between 0.000 and 0.999 inclusive, correct to 3 significant figures. She plans to use her calculator to choose a sample of members from the 851 members in her health club. She numbers the members from 1 to 851. Then she uses her calculator to generate some random numbers. She multiplies each random number by 851 and rounds up to the next whole number to give the number of a member in the sample. This is called a 'member number'.
Find all possible random numbers, correct to 3 decimal places, that would produce the following member numbers.
A member number of 680 .
A member number of 850 .
Explain briefly how your answers to part (b) show that Maroulla's method does not produce a random sample.
The times, in minutes, taken by students to complete a test have mean and standard deviation . The times taken by a random sample of 100 students are noted and are used to calculate a 95% confidence interval for .
Given that the end points of the 95% confidence interval are 31.02 and 33.98, correct to 4 significant figures, calculate the value of .
The calculation of the confidence interval required the use of the Central Limit theorem.
Explain why it is valid to use the Central Limit theorem in this case.
A researcher calculates a number, r, of 95\% confidence interval for .
Find the largest value of r such that the probability that all r confidence intervals contain the true value of is greater than 0.5.
A population is normally distributed with mean 35 and standard deviation 8.1. A random sample of size 140 is chosen from this population and the sample mean is denoted by .
Find .
It is given that . Find the value of a.
