Finding Probabilities
`The random variable `X~B(8,1/3)`. Find `P(X=2)
0.273
A balloon manufacturer claims that 95% of his balloons will not burst when blown up. If you have 20 of these balloons to blow up for a birthday party, what is the probability that none of them burst when blown up?
0.358
Explain what is meant by a critical region.
The critical region is a region of the probability distribution which, if the test statistic falls within it, would cause you to reject the null hypothesis.
`A random variable has distribution `X~B(10,p)`. A single observation of `x=1` is taken from this distribution. Test, at the 5% significance level, `H_0:p=0.4` against `H_1:p<0.4`.
0.0464<0.05 `. There is sufficient evidence to reject `H_0` so `p<0.4.
`A random variable has distribution `X~B(25,p)`. A single observation of `x=10` is taken from this distribution. Test, at the 10% significance level, `H_0:p=0.3` against `H_1:p!=0.3`.
P(X>=10)=0.189>0.05`. There is insufficient evidence to reject `H_0` so there is no reason to doubt that `p=0.3`.
`The random variable `T~B(15,2/3)`. Find `P(3<=T<=4)
0.00178
The probability of a six is constant (0.3). There are two outcomes (six and not six). There is a fixed number of trials (15). Each trial is independent.
X~B(15,0.3)
A mechanical component fails, on average, 3 times out of every 10. An engineer designs a new system of manufacture that he believes reduces the likelihood of failure. He tests a sample of 20 components made using his new system.
Describe the test statistic, and state the hypotheses.
`The test statistic is the number of times the sample fails. `H_0: p=0.3, H_1:p<0.3
`A random variable has distribution `X~B(8,p)`. A single observation of `x=7` is taken from this distribution. Test, at the 5% significance level, `H_0:p=0.32` against `H_1:p>0.32`.
0.0020<0.05`. There is sufficient evidence to reject `H_0` so `p>0.32
A coin is tossed 20 times and lands on heads 6 times. Use a two-tailed test with a 5% significance level to determine whether there is sufficient evidence to conclude that the coin is biased.
P(X<=6)=0.0577>0.025. X=6 ` does not lie in the critical region so there is no reason to think that the coin is biased.
`The random variable `X~B(20,0.35)`. Find `P(X>6)
0.5834
In a town, 30% of residents listen to the local radio. Ten residents are chosen at random. X = the number of these 10 residents that listen to the local radio. Find the probability that at least half of these 10 residents listen to local radio.
0.1503
`A test statistic has a distribution `B(10,p)`. Given that `H_0: p=0.2, H_1:p>0.2`, find the critical region for the test using a 5% significance level.
`The critical region is `X>=5`. (Since `P(X>=5)=0.0328<0.05`)
The success rate of the standard treatment for patients suffering from a particular skin disease is claimed to be 68%. A random sample of 10 patients receives the standard treatment and in only 3 cases was the treatment successful. It is thought that the standard treatment is not as effective as it is claimed. Test the claim at the 5% significance level.
P(X<=3)=0.0155<0.05`. There is sufficient evidence to reject the null hypothesis so `P<0.68`. The treatment is not as effective as claimed.
A machine makes glass bowls and it is observed that one in ten of the bowls have hairline cracks in them. The production process is modified and a sample of 20 bowls is taken. 1 of the bowls is cracked. Test, at the 10% level of significance, the hypothesis that the proportion of cracked bowls has changed as a result of the change in the production process.
`Test statistic: the number of cracked bowls. `H_0: p=0.1, H_1:p!=0.1. P(X<=1)=0.3917=39.17%>5%` so there is not enough evidence to reject `H_0`. The proportion of cracked bowls has not changed.
`The random variable `X~B(40,0.47)`. Find `P(10<X<17)
0.2302
A six-sided die is biased. When the die is thrown the number 5 is twice as likely to appear as any other number. All the other faces are equally likely to appear. The die is thrown repeatedly. Find the probability that the first 5 will occur on the sixth throw.
0.0531
`A random variable has distribution `B(10,p)`. A single observation is used to test `H_0:p=0.22` against `H_1: p!=0.22`. Using a 1% level of significance, find the critical region of this test. The probability in each tail should be as close as possible to 0.005. Write down the actual significance level of the test.
`Critical region is `X=0` and `7<=X<=10`. The actual significance level is 8.5%
A marketing company claims that Chestly cheddar cheese tastes better than Cumnauld cheddar cheese. Five people chosen at random as they entered a supermarket were asked to say which they preferred. Four people preferred Chestly. Test, at the 5% level of significance, whether or not the manufacturer's claim is true.
H_0: p=0.5, H_1:p>0.5. P(X>=4)=0.1875>0.05. `There is insufficient evidence to reject `H_0`. There is insufficient evidence that the manufacturers claim is true.
A standard blood test is able to diagnose a particular disease with probability 0.96. A manufacturer suggests that a cheaper test will have the same probability of success. It conducts a clinical trial on 75 patients. The new test correctly diagnoses 63 of these patients. Test the manufacturer's claim at the 10% level.
`Test statistic: the number of patients correctly diagnosed. `H_0: p=0.96, H_1:p!=0.96. P(X<=63)=0.0000417<0.05` so there is enough evidence to reject `H_0`. The new test does not have the same probability of success as the old test.
`The random variable `X~B(50,0.4)`. Find the smallest number `r` such that `P(X>r)<0.01
r=28
A factory produces a component for the motor trade and 5% of the components are defective. A quality control officer regularly inspects a random sample of 50 components.
The officer will stop production if the number of defectives in the sample is greater than a certain value, d. Given that the officer stops production less than 5% of the time, find the smallest value of d.
d=5
A restaurant owner notices that her customers typically choose lasagne one fifth of the time. She changes the recipe and believes this will change the proportion of customers choosing lasagne.
She takes a random sample of 25 customers. Find at the 5% level of significance, the critical region for a test to check her belief. State the probability of incorrectly rejecting the null hypothesis.
`The critical region is `X<=1` and `X>=10`. The probability of incorrectly rejecting `H_0` is 4.47%.
A polling organisation claims that the support for a particular candidate is 35%. It is revealed that the candidate will pledge to support local charities if elected. The polling organisation think that the level of support will go up as a result. It takes a new poll of 50 voters. Using a 5% significance level, find the critical region for a test to check the belief.
`Critical region is `X>=24.
From the large data set, the likelihood of a day with either zero or trace amounts of rain in Hurn in June 1987 was 0.5. Poppy believes that the likelihood of a rain-free day in 2015 has changed. In June 2015 in Hurn, 21 days were observed as having zero or trace amounts of rain. Using a 5% significance level, test whether or not this is evidence to support Poppy's claim.
P(X>=21)=0.021<0.025`. Therefore there is sufficient evidence to support Poppy's claim that the likelihood of a rain-free day has changed.