<p>The security guard of a particular school claims that the average speed of the cars in the school compound is greater than the speed limit of $25$ km/h. To investigate the security guard’s claim, the traffic police randomly selected $50$ cars and the speed was recorded. The total speed and the standard deviation of the $50$ cars are found to be $1325$ km/h and $7.75$ km/h respectively.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Find the unbiased estimates of the population mean and variance.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Test at $5$% level of significance whether there is sufficient evidence to support the security guard’s claim.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Without any further test, state the conclusion if instead the traffic police is testing that the average speed of the cars is not $25$ km/h at $20$% level of significance</div><p style="text-align: right;">[2]</p></li></ol>

The security guard of a particular school claims that the average speed of the cars in the school compound is greater than the speed limit of $25$ km/h. To investigate the security guard’s claim, the traffic police randomly selected $50$ cars and the speed was recorded. The total speed and the standard deviation of the $50$ cars are found to be $1325$ km/h and $7.75$ km/h respectively.

Find the unbiased estimates of the population mean and variance.

Test at $5$% level of significance whether there is sufficient evidence to support the security guard’s claim.

Without any further test, state the conclusion if instead the traffic police is testing that the average speed of the cars is not $25$ km/h at $20$% level of significance

Default solution

$\bar{x} = \dfrac{1325}{50} = 26.5$, $\sigma_x = 7.75$

$s^2 = \dfrac{n}{n-1}\,\sigma_x^{\,2} = \dfrac{50}{49}(7.75)^2 = 61.288 = 61.3$ (3 s.f.)

Let $X$ be the speed of a car in the school compound.

$H_0: \mu = 25$ km/h
$H_1: \mu > 25$ km/h

Under $H_0$, test statistic, $Z = \dfrac{\bar{X} - \mu}{\sqrt{s^2/n}} \sim \text{N}(0, 1)$, approximately by Central Limit Theorem since $n = 50$ is large.

$n = 50$, $\bar{x} = 26.5$, $s^2 = 61.288$, $\alpha\% = 5\%$

From GC, p-value $= 0.0877 > 5\%$

Do not reject $H_0$. At 5% level of significance, we do not have sufficient evidence to state that the mean speed of cars has increased.

$H_0: \mu = 25$ km/h
$H_1: \mu \neq 25$ km/h

**Method 1 (p-value):**
New p-value $= 2 \times 0.0877 = 0.1754 < 20\%$

$\therefore$ Reject $H_0$.

**Method 2:**
Old p-value $= 0.0877 < \dfrac{20\%}{2} = 10\%$

$\therefore$ Reject $H_0$. At 20% level of significance, the average car speed is not 25 km/h.

2019 CJC P2 Q9 [Modified]

2019 CJC P2 Q9 [Modified]

Video Solution:

Default solution