June 2006 MS – S3 OCR.pdf

Mark Scheme 4734, June 2006, covers various statistical concepts, including Poisson distributions, hypothesis testing, and confidence intervals.

1. The first question involves adding two Poisson distributions with a mean of 17, and the answer is P(27) = e^(-17) * (17^27) / 27!, which equals 0.00634.

2. The second question tests the equality of proportions, with the null hypothesis (H0) stating that all proportions are equal and the alternative hypothesis (H1) stating that they are not. The expected values under H0 are calculated to be 150, and the chi-squared statistic (X2) is found to be 8.653, which is greater than the critical value of 7.815, leading to the rejection of H0.

3. The third question assumes a normal distribution for the population of differences and calculates the t-statistic to determine if there is evidence that the mean time has reduced. The t-statistic is found to be 2.520, which is greater than the critical value, leading to the rejection of the null hypothesis.

4. The fourth question involves solving an equation and finding the value of q3, which is calculated to be √10.

5. The fifth question covers expected values, with E(X2) and E(X) calculated using integrals. The value of 'a' is then calculated as the ratio of E(X2) to E(X), which equals 2.6659.

6. The sixth question tests the independence of volume and day, with the null hypothesis stating that they are independent and the alternative hypothesis stating that they are not. The test statistic is found to be greater than the critical value, leading to the rejection of the null hypothesis.

7. The seventh question involves normal distributions, with T1 and T2 following normal distributions. The probability that T2 is less than half of T1 is calculated to be 0.189.

8. The eighth question covers hypothesis testing, with the null hypothesis stating that the mean of population B is equal to the mean of population A, and the alternative hypothesis stating that the mean of population B is greater than the mean of population A. The z-statistic is calculated to be 1.700, which is greater than the critical value, leading to the rejection of the null hypothesis.