You are asked to find the length of the longest day in Galway.

(i)The 21st of June is the longest day which is 92 days after March 21st .

We are given f(t)= 12.25+4.75Sin(2pi(t))/365 ) We just need to find f(92) =17 hours

(ii)My favourite way is based on the range of the periodic function a+bSin(kx) The range of this function is a-b to a+b

The range of 12.25+4.75Sin(2pi(t))/365) is 12.25-4.75 to 12.25+4.75 therefore the longest day is 12.25+4.75 = 17 hours.

Of course you can also use f’(t) = 0!

There are 7 situations you need to know when dealing with inferential statistics .

(I)Probability from a normal distribution

(ii)The test statistic from a sample

(iii)Conclusions drawn from test statistic If -1.96 < z< 1.96 the result is not significant at the 5% level

(iv)The 95% confidence interval for the mean of a population based on a sample x**±1.96( σ/√n) .**

(v)The test statistic for a population proportion (pˆ -Ρ)/(√PQ/n) ,P is the population proportion ,Q = 1-P .

(vi)The 95% confidence interval for a proportion pˆ **±1.96 √((pˆqˆ)/n) **

(vii)The P value is the probability that the data is compatible with the null hypothesis Ηο

**To find the P value ( -1.96>P(z) or P(z)>1.96)**

(i)Find the test statistic (z) (ii)Find P(z) from the tables .(iii)Find 1 -P(z) (iv)Multiply 1 -P(z) by 2 .

If the result is < 0.05 we do not accept the null hypothesis “if the P is low the null must go ”

If the result is >0.05 We accept the null hypothesis.” if the P is high the null must fly”

For a free set of worked solutions on inferential statistics.

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