JC Math

Worked Example

Find the range of values of x for which the series 1 + (2x − 1) + (2x − 1)² + … is convergent, and find the sum to infinity in terms of x.

Many students write 2x − 1 < 1 and give x < 1, but x = −0.5 gives r = −2, and that series does not converge. The condition is |2x − 1| < 1.

r = 2x − 1, and the series converges when |r| < 1

−1 < 2x − 1 < 1, so 0 < x < 1 (not x < 1)

S_∞ = 11 − (2x − 1) = 12(1 − x)

How we help: We teach you to check the conditions a formula needs before you apply it. Here, the sum-to-infinity formula needs |r| < 1, so you solve |2x − 1| < 1 first.

Worked Example

The mean waiting time at a clinic was 12 minutes. After a change in procedure, a random sample of 50 patients has a mean waiting time of 12.8 minutes and an unbiased estimate of the population variance of 9.0. Test, at the 5% level of significance, whether the mean waiting time has increased.

Many students work out the test correctly and stop at “reject H₀”. The conclusion has to say what that means for the waiting time, at the 5% level.

H₀: μ = 12 against H₁: μ > 12

z = 12.8 − 12√(9.0 ÷ 50) = 1.886, so p = 0.0297 < 0.05

Reject H₀: there is sufficient evidence at the 5% level that the mean waiting time has increased

How we help: We teach you to write up a hypothesis test in full: the hypotheses, the test statistic and a conclusion in context. Here, “reject H₀” becomes a statement about the mean waiting time, at the 5% level.