3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 . After having the series of lessons with David, Becki decides to investigate his claim.
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, \(x\), that she takes to complete the round. Her results are summarised below, where \(\bar { x }\) denotes the sample mean. $$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$ Using a \(z\)-test and the \(1 \%\) level of significance, investigate David's claim.

# Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $H_0: \mu = 36$; $H_1: \mu < 36$ | B1 | |
| $\bar{x} = \frac{1730}{50} = 34.6$ | B1 | |
| $s^2 = \frac{784}{49} = 16$ | B1 | |
| Test statistic: $z = \frac{34.6 - 36}{\frac{4}{\sqrt{50}}} = -2.47$ | M1, A1 | $-2.48$ to $-2.47$ |
| $z_{\text{crit}} = -2.3263$ | B1 | |
| Reject $H_0$ | $A1\checkmark$ | |
| Sufficient evidence at 1% level of significance to support David's claim | $E1\checkmark$ | |

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3 David is the professional coach at the golf club where Becki is a member. He claims that, after having a series of lessons with him, the mean number of putts that Becki takes per round of golf will reduce from her present mean of 36 .

After having the series of lessons with David, Becki decides to investigate his claim.\\
She therefore records, for each of a random sample of 50 rounds of golf, the number of putts, $x$, that she takes to complete the round. Her results are summarised below, where $\bar { x }$ denotes the sample mean.

$$\sum x = 1730 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 784$$

Using a $z$-test and the $1 \%$ level of significance, investigate David's claim.

\hfill \mbox{\textit{AQA S2 2007 Q3 [8]}}