An industrial chemist is studying the relationship between the UV intensity index, uuu, and the yield of a refined polymer, yyy grams. The chemist records the results for 12 different batches. The summary statistics are as follows:
∑u=84,∑y=156,∑u2=628,∑y2=2118 \sum u = 84, \quad \sum y = 156, \quad \sum u^2 = 628, \quad \sum y^2 = 2118 ∑u=84,∑y=156,∑u2=628,∑y2=2118 ∑uy=1146,Suu=40,Syy=90 \sum uy = 1146, \quad S_{uu} = 40, \quad S_{yy} = 90 ∑uy=1146,Suu=40,Syy=90Find the mean and variance of the recorded yields.
The chemist believes that increased UV intensity results in a higher polymer yield.
(i) Calculate the product moment correlation coefficient (PMCC) between uuu and yyy. (ii) State, with a reason, whether or not this value supports the chemist's belief.
Find the least squares regression line of yyy on uuu in the form y=a+buy = a + buy=a+bu. Give the values of aaa and bbb to 3 significant figures.
Estimate the yield when the UV intensity index is 9.2.
The coding z=5y−50z = 5y - 50z=5y−50 is used to transform the yield data.
Find an equation of the least squares regression line of zzz on uuu in the form z=c+duz = c + duz=c+du.
Find: (i) the variance of the transformed yields zzz. (ii) the product moment correlation coefficient between zzz and uuu.
316 exam-style questions on OCR A Level Maths 2.5 Statistical Hypothesis Testing, covering 2.5.1 The language of hypothesis testing, 2.5.2 Hypothesis test for a binomial proportion, 2.5.3 Inference and significance level, 2.5.4 Sample mean as a random variable (A-level only), 2.5.5 Hypothesis test for the mean of a normal distribution (A-level only), 2.5.6 Pearson's product-moment correlation coefficient, and 2.5.7 Hypothesis test using Pearson's coefficient (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.