The treadwear index provided on car tyres helps prospective buyers make their purchasing decisions by indicating a tyre’s resistance to tread wear. A tyre with a treadwear grade of 200 should last twice as long, on average, as a tyre with a grade of 100. A consumer advocacy organisation wishes to test the validity of a popular branded tyre that claims a treadwear grade of 200. A random sample of 24 tyres indicates a sample mean treadwear index of 195.4 and a sample standard deviation of 19.8.   (a) Using 0.05 level of significance, is there evidence to conclude that the tyres are not meeting the expectation of lasting at least twice as long as a tyre graded at 100? Show all your workings.  (b) What assumptions are made in order to conduct the hypothesis test in (a)?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...