Mathematics & Statistics Statistics Examples

HT test, Q2 It is known that 20% of all farms in a particular county exceed 160 acres and that 60% of all farms in that county are owned by persons over 50 years old. Of all farms in the county exceeding 160 acres, 55% are owned by persons over 50 years old. What is the probability that a randomly chosen farm in this county both exceeds 160 acres and is owned by a person over 50 years old? What is the probability that a farm in this county either is bigger than 160 acres or is owned by a person older than 50 years old (or both)? What is the probability that a farm in this county, owned by a person older than 50 years old, exceeds 160 acres? Are size of farm and age of owner in this county independent?

HT Test, Q3 It is estimated that 45% of graduates from a particular university graduate with a II.1 or a I (in the UK these are called “good degrees”). Let X denote the number of graduates with a “good degree” from a random sample of 7 students. What distribution does X follow? Compute the probability that at most two students graduate with a good degree. In a particular year, 123 students graduate from the university. Find the mean and standard deviation of the number of graduates with a good degree. Use a normal approximation to compute the probability that a majority of these 123 graduates obtain a good degree.

HT Test, Q4 Betty’s is the most famous tearoom in York and they sell tea in boxes of (nominally) 100 grams. Because of EU legislation they take extra care and wrap, on average, 101.0 grams in each box. Assume that the weight of the boxes is normally distributed with a standard deviation of 0.6 grams. What is the probability that a box does not contain enough tea (round to two decimals)? Betty’s want to reduce the probability found in (a) by half by reducing the standard deviation. This will be achieved by making the wrapping machine more precise. What standard deviation should Betty’s aim for to achieve its goal?

HT Test, Q5 A random sample of Irish workers has been obtained by asking 20 Dublin Bus drivers outside Ringsend terminal about their wages. We found an average gross wage of 22,350 Euros, with a standard deviation of 3,200 Euros. In order to obtain some feeling about the accurateness of this average, we computed a 95% confidence interval around the mean. We do not have evidence on the exact shape of the income distribution, but the Central Limit Theorem tells us that the sampling distribution of ­the mean is approximately normal with mean μ and standard deviation [(3,200)2/20]1/2=715.54. Using the formula μ±zα/2s/√n, we find a 95% confidence interval of [22350-1.64*715.54,22350+1.64*715.54] =[21176.51,23523.49]. So, 95% of Irish workers will earn between 21,177 and 23,523 Euros.

Sampling distributions Suppose that 50% of all Irish adults believe that a major overhaul of the HSE is essential. What is the probability that more than 56% of a random sample of 150 Irish adults would hold this belief?

Estimation A clothing store is interested in how much 3rd level students spend on clothing during the first month of the academic year. For a random sample of nine students the mean expenditure was €157.82, and the sample standard deviation was €38.89. Assume that the population is normal. Find a 95% confidence interval for the population mean. What is the margin of error of this interval?

Hypothesis tests In a random sample of 545 accountants engaged in preparing company financial reports, 117 indicated that estimates of the cash flow were the most difficult element of the budget to derive. Test the null hypothesis that at least 25% of all accountants find cash flow estimates the most difficult estimates to derive at the 5% significance level. What does the significance level represent?