The Post Office in a certain country claims that 80% of letters posted are delivered the next day. Consumer Council believes that the percentage is less than this. To investigate the claim, it arranges for 50 letters to be posted and it counts the number delivered the next day.
1.state the appropriate hypothesis.
2.in the event, 34 of the letters are delivered the next day. calculate the
p-value of this result and interpret in context.
3.the consumer council sets up a larger investigation in which 1000 letters are posted and their delivery is monitored. It is found that 779 are delivered the next day. Calculate an approximate p-value of this result and interpret in its context.
h0 : (mean)=50
h1 : (mean)=/=50
But I don't know if this is asking me to use CLT, which distribution am I going to use and the n(sample mean) is...
I don't think you've got the null and alternative hypotheses right, and I'm pretty sure you don't apply the Central Limit Theorem here.
I'd say if you have a problem determining the distributions in questions, just think about what they require in order to be modelled with a certain one. For example, the Binomial distribution requires that there be two outcomes, success or failure, and fulfil a couple other criteria; the Poisson distribution requires that there be a mean rate of occurrences etc.
This one should be modelled by a Binomial distribution as it fits the criteria nicely.
This is my thought process behind the question:
1) H0: p = 0.8; H1: p < 0.8
2) X~B(50,0.8) -> Y~B(50,0.2) (as 0.8 not in tables)
P(X<=34) = P(Y=>16) = 1-P(Y<=15) = 0.0308.
Depending on the significance level (generally it's 5%), this would mean that the result would be significant enough to reject the null hypothesis as the p-value found is smaller than 0.05. This would mean that the Consumer Council are correct in believing that the percentage of letters delivered the next day by the Post Office is less than 0.8.
If the significance level was smaller than 0.0308 (e.g. 2.5% level), then it wouldn't be a significant enough value to reject the null hypothesis that the Post Office deliver 80% of post the next day.
3) Now n = 1000, and we still assume p = 0.8
H0: p = 0.8; H1: p < 0.8
Convert this to a normal distribution as n is very large and p is around the 0.5 mark. Therefore the mean is 1000 x 0.8 which = 800, and variance = 800(1-0.8) = 160
P(X<=779) (using continuity correction) -> P(Y<779.5)
P(Z<(779.5 - 800)/4root10) = P(Z<-1.62) = 1 - P(Z<1.62) = 1 - 0.9474 = 0.0526
Again, this value's significance depends on the significance level of the experiment. If the significance level is 5% or below, then the Post Office's claim of delivering 80% of mail the next day is correct, so we accept H0. If the significance level is above that then H0 is rejected.
Hope I helped!
Mate, thank you very much
I've missed the lesson that I should be learning these testings, notes were given but no examples there
thanks for solving this for me and the explanation is detailed
really, thanks a lot.