Statistical hypothesis testing

For 10 days, a café recorded the maximum temperature, x °C, and the number of cold drinks sold, y. The temperatures ranged from 10 °C to 30 °C. The regression line of y on x is y = 12.5 + 3.2x and the product moment correlation coefficient is 0.87. (a) Interpret the value 3.2 in context. (b) Estimate the number of cold drinks sold on a day with a maximum temperature of 25 °C, and comment on the reliability of your estimate. (c) Explain why the line should not be used to estimate sales when x = 40. (d) Test, at the 5% level, whether there is evidence of positive correlation. (One-tailed critical value for n = 10: 0.5494.)

Mo suspects that a six-sided dice is biased. He rolls it 60 times and gets 17 sixes. (a) Find the critical region for a two-tailed test, at the 5% level, of whether the probability of a six differs from 1/6. Use 2.5% in each tail. (b) State the actual significance level of the test. (c) Carry out the test, stating your conclusion in context.

The lengths of bolts made by a machine are normally distributed with standard deviation 3 mm. The mean length should be 25 mm. A random sample of 16 bolts has mean length 26.4 mm. (a) Test, at the 1% level of significance, whether the mean length differs from 25 mm. (b) Find the critical region for the sample mean in this test. (c) Explain why it is important that the sample is random.