A Level Maths Correlation Hypothesis Testing questions

A teacher records, for a random sample of 12 students, the number of hours spent revising and the mark in a mock exam. The product moment correlation coefficient is \(r = 0.62\). The critical value for a one-tailed test at the 5% level with \(n = 12\) is 0.4973.

(a) Test, at the 5% level of significance, whether there is evidence of positive correlation between revision time and mock mark. (3)

A sample of 30 students gives a PMCC of 0.29 between hours of sleep the night before an exam and exam score. The critical value for a one-tailed test at the 5% level is 0.3061.

(a) Write down suitable hypotheses for a test of whether more sleep is associated with higher scores. (2)

(b) State the conclusion of the test. (1)

For a random sample of 20 towns, the product moment correlation coefficient between the average house price and the percentage of residents aged over 65 is \(-0.41\). For \(n = 20\), the critical values for the PMCC are: 5% one-tailed 0.3783; 2.5% one-tailed 0.4438; 1% one-tailed 0.5155.

(a) Test, at the 5% level of significance, whether there is any correlation between these two variables. (4)

(b) State, with a reason, how your conclusion would differ if the test had been for negative correlation at the 5% level. (1)

A garden centre records, for 15 randomly chosen days, the maximum temperature and the number of bags of compost sold. The PMCC is 0.48 and software gives the \(p\)-value for a test of \(H_0: \rho = 0\) against \(H_1: \rho \gt 0\) as 0.0351.

(a) Test at the 5% level of significance whether there is positive correlation. (2)

(b) State the conclusion if the test had been carried out at the 1% level. (1)

(c) The manager says "higher temperatures cause more compost to be sold". Comment on this statement. (1)

An ecologist believes that the more cars that pass a hedge each hour, the fewer species of bird nest in it. She collects data from 10 randomly chosen hedgerows and finds that the PMCC between hourly traffic count and number of nesting species is \(-0.58\). Critical values for \(n = 10\): 5% one-tailed 0.5494, 2.5% one-tailed 0.6319.

(a) Explain why a one-tailed test is appropriate. (1)

(b) Test, at the 2.5% level, whether there is evidence to support her belief. (4)

(c) State whether the conclusion would change at the 5% level. (1)

A physics student measures the height, \(h\) m, of a ball thrown upwards at 14 equally spaced times \(t\) s from launch until it lands. She finds the PMCC between \(t\) and \(h\) is 0.04. The critical value for a two-tailed test at the 5% level with \(n = 14\) is 0.5324.

(a) Test at the 5% level whether there is correlation between \(t\) and \(h\). (3)

(b) The student concludes that height does not depend on time. Explain why she is wrong. (1)

For a random sample of 25 car journeys, the correlation coefficient between average speed and fuel consumption (litres per 100 km) is \(-0.38\). The \(p\)-value for a two-tailed test of \(H_0: \rho = 0\) is 0.0610.

(a) Investigate, at the 10% level of significance, whether there is a linear correlation between average speed and fuel consumption. (3)

(b) Write down the \(p\)-value for a one-tailed test of \(H_0: \rho = 0\) against \(H_1: \rho \lt 0\). (1)

A biologist records the number, \(N\), of bacteria in a culture at time \(t\) hours on 8 occasions. She calculates \(y = \log_{10} N\). The PMCC between \(t\) and \(y\) is 0.97 and the equation of the regression line of \(y\) on \(t\) is \(y = 0.31 + 0.12t\). The critical value for a one-tailed test at the 1% level with \(n = 8\) is 0.7887.

(a) Test, at the 1% level, whether there is positive correlation between \(t\) and \(y\). (3)

(b) Show that the model can be written as \(N = ab^t\), giving the values of \(a\) and \(b\) to 3 significant figures. (3)

(c) Interpret the value of \(b\) in context. (1)

A market researcher expects the PMCC between two variables to be about 0.35. Part of a table of critical values for a one-tailed test at the 5% level is given below.

(a) Assuming the sample PMCC is 0.35, find the smallest sample size in the table for which there would be evidence, at the 5% level, of positive correlation. (2)

(b) Explain why the critical value decreases as \(n\) increases. (1)

A dietitian records the daily sugar intake and the body mass index of 11 randomly chosen adults. The PMCC is 0.21. One adult's data point is found to be a recording error and is removed; the PMCC for the remaining 10 adults is 0.74. One-tailed 5% critical values: \(n=11\): 0.5214; \(n=10\): 0.5494.

(a) Test, at the 5% level, for positive correlation using (i) all 11 adults, (ii) the 10 remaining adults. (4)

(b) Comment on the effect of the error on the analysis. (1)

An electricity supplier collects data from 12 homes on floor area, \(x\) m², and mean monthly electricity cost, \(£y\). The values of \(x\) range from 55 to 160. The regression line of \(y\) on \(x\) is \(y = 4.2 + 0.85x\) and the PMCC is 0.91. The critical value for a one-tailed test at the 0.5% level with \(n = 12\) is 0.7079.

(a) Test, at the 0.5% significance level, whether there is positive correlation between floor area and monthly cost. (3)

(b) Interpret the gradient of the regression line. (1)

(c) Estimate the monthly cost for a flat of floor area 30 m², and comment on the reliability of your estimate. (2)

Two researchers each test \(H_0: \rho = 0\) against \(H_1: \rho \gt 0\) using the same sample of 18 pairs of data. Researcher A uses a 5% significance level and Researcher B uses a 1% significance level. For \(n = 18\) the one-tailed critical values are 0.4000 (5%) and 0.5425 (1%).

(a) Find the range of values of the sample PMCC, \(r\), for which the researchers reach different conclusions. (2)

(b) Researcher B says "Using 1% means I am less likely to wrongly claim there is correlation." Explain whether she is correct. (1)