Practise Probability & Statistics 2 through Poisson and normal models, continuous variables, sampling, confidence intervals and hypothesis tests with contextual conclusions.
The random variables X and Y have independent distributions and respectively.
Find P(2<X<5).
Find P(X+Y>2).
The total of 100 random values of X and 150 random values of Y is denoted by T.
Use a suitable approximating distribution to find P(T<560).
The random variable W has a Poisson distribution.
State the relationship between E(W) and .
The random variable X has the distribution B(n, p). Jyothi wishes to use a Poisson distribution as an approximate distribution for X.
Use the formulae for E(X) and to explain why it is necessary for p to be close to 0 for this to be a reasonable approximation.
Given that Y has the distribution B(20000,0.00007), use a Poisson distribution to calculate an estimate of P(Y>2).
A researcher is investigating whether the proportion of families who do not own a car in his town is different from the proportion of the population in the whole country, which is 10.1%. He takes a large random sample of families in his town and finds the proportion of families that do not own a car.
Explain why a two-tailed test is appropriate in this context.
State suitable null and alternative hypotheses for the test.
The researcher calculates the value of the test statistic z and finds that z=1.82. He carries out the test at the 5\% significance level.
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 .
Test, at the 2% significance level, whether type B batteries have a shorter mean life than type A batteries.
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.
A certain website receives an average of hits per hour. In the past the value of was 14.4. After making some improvements, the owner of the website wishes to test whether the value of has increased. He chooses a 10 -minute period at random and finds that there were 6 hits during this period. You may assume that the number of hits the website receives in any given time period follows a Poisson distribution.
Carry out the test at the 2.5% significance level.
Explain whether it is possible that a Type I error or a Type II error or both may have been made in carrying out the test.
A sports fan produces a magazine each month.
On average 1 in 540 characters in the magazine is incorrect.
Use an appropriate approximating distribution to find the probability that, in a magazine containing 2430 characters, there are at least 4 incorrect characters.
Justify your approximating distribution.
On average the number of copies, X, of the magazine sold per month is 123.4.
State one condition for X to have a Poisson distribution.
You are now given that X has a Poisson distribution.
Use an appropriate approximating distribution to find the probability that in a randomly chosen month, more than 130 copies of the magazine are sold.
The masses, in kilograms, of large and small bags of potatoes have the independent distributions N(2.5,0.05) and N(0.8,0.02) respectively.
Find the probability that the total mass of a randomly chosen large bag of potatoes and a randomly chosen small bag of potatoes is more than 3.55 kg .
Find the probability that the mass of a randomly chosen large bag of potatoes is less than 3 times the mass of a randomly chosen small bag of potatoes.
The time, in minutes, taken by students to complete a test is modelled by the random variable X with probability density function
Find the probability that a randomly chosen student takes longer than 4.5 minutes to complete the test.
Write down the median of X.
Without performing an integration, use your answer to part (a) to find P(3.5<X<4.5).
The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a straight line from ( 0, a ) to ( b, 0 ) where a and b are constants. Elsewhere f(x)=0.
Find an expression for b in terms of a.
Given that find the value of a.
（c）Using the value of a found in part（b）find the value of k such that ．
