<ol class="abc-list" start="2"><li><div class="list-item-paragraph">In general, the marks obtained by students in their Mathematics examination may be assumed to be a random variable with mean $65$ and variance $32$. After the introduction of a new program of revision classes, it is found that there are improvements to the marks of the some of the students.</div><div class="list-item-paragraph">A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.<span class="inline-marks">[3]</span></div></li><li><div class="list-item-paragraph">A sample of $n$ observations is taken form a population with mean $5$. The sample mean is found to be $5.1$ and sample standard deviation $0.7$. A 2-tailed normal test ($z$-test) is conducted and the null hypothesis is rejected at the $5\%$ significance level. Find the least value of $n$.</div></li></ol>

In general, the marks obtained by students in their Mathematics examination may be assumed to be a random variable with mean $65$ and variance $32$. After the introduction of a new program of revision classes, it is found that there are improvements to the marks of the some of the students.

A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.A random sample of the marks of $50$ students is taken and the mean is found to be $66.4$. A test is conducted and it shows that there is insufficient evidence that the new program of revision classes is effective. Find an inequality satisfied by the significance level of the test.

A sample of $n$ observations is taken form a population with mean $5$. The sample mean is found to be $5.1$ and sample standard deviation $0.7$. A 2-tailed normal test ($z$-test) is conducted and the null hypothesis is rejected at the $5\%$ significance level. Find the least value of $n$.

Hypothesis Testing - General

Hypothesis Testing

International Baccalaureate Higher Level Mathematics

IB HL Math

2007 IJC P2 Q11

$E(X) = 65$, $\text{Var}(X) = 32$

Let $X$ be the marks obtained by a student.
Let $\mu$ be the population mean marks obtained by a student.

$H_0: \mu = 65$
$H_1: \mu > 65$

$\bar{x} = 66.4$, $n = 50$, $\sigma^2 = 32$

Under $H_0$, test statistic $Z = \dfrac{\bar{X} - 65}{\sqrt{\dfrac{32}{50}}} \sim N(0, 1)$ approximately by Central Limit Theorem.

From GC, $p\text{-value} = 0.0401$

Since $H_0$ is not rejected:
$p\text{-value} > \alpha\%$
$\alpha\% < 0.0401$
$\alpha < 4.01$

Level of significance is lesser than 4.01%.

$\bar{x} = 5.1$, $\sigma_x = 0.7$, $\alpha\% = 5\%$, $n = ?$

$H_0: \mu = 5$
$H_1: \mu \neq 5$

$s^2 = \dfrac{n}{n-1}\sigma_x^2$, so $\dfrac{s^2}{n} = \dfrac{\sigma_x^2}{n-1} = \dfrac{0.7^2}{n-1}$

$H_0$ is rejected. Assume that $n$ is large.

Under $H_0$, test statistic $Z = \dfrac{\bar{X} - 5}{\sqrt{\dfrac{s^2}{n}}} \sim N(0, 1)$ approximately by Central Limit Theorem.

Two-tailed test at 5%: $z_{\text{critical}} = \pm 1.95996$

Since $H_0$ is rejected and $\bar{x} = 5.1 > 5$:
\[z_{\text{stat}} \geq 1.95996\]
\[\dfrac{5.1 - 5}{\sqrt{\dfrac{0.7^2}{n-1}}} \geq 1.95996\]
\[\dfrac{0.1}{0.7}\sqrt{n-1} \geq 1.95996\]
\[\sqrt{n-1} \geq 7(1.95996)\]
\[n - 1 \geq 188.23\]
\[n \geq 189.23\]

Least $n$ is 190.

2007 IJC P2 Q11

2007 IJC P2 Q11

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