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PSLE Foundation Mathematics

<p>A gaming company claims that gamers spend an average of $m$ hours daily on a new game that they have launched. A random sample of $30$ gamers was surveyed and the time, $x$ hours, spent daily on the game was recorded. The results are summarized by</p>
<p style="text-align: center">$\sum{x}=126$ , $\sum{{{x}^{2}}}=550.1525$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Calculate the unbiased estimates of the population mean and variance of the number of hours a gamer spends daily on the game.</div><p style="text-align: right;">[2]</p></li></ol>
<p>A test is carried out, at the $5\%$ significance level, to determine whether the average time spent daily on the game has been overstated by the company.</p>
<ol class="i-list" start="2"><li><div class="list-item-paragraph">State appropriate hypotheses for the test.</div><p style="text-align: right;">[1]</p></li><li><div class="list-item-paragraph">Find the range of values of $m$ for which the null hypothesis is not rejected.</div><p style="text-align: right;">[3]</p></li></ol>
<p>A second sample of $50$ gamers is surveyed and the time, $y$ hours, spent daily on the game was recorded, and the results are summarized by</p>
<p style="text-align: center">$\sum{y}=215$,$\sum{{{y}^{2}}}=964.19$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">The two samples are combined into a single sample and a test is carried out at the $5\%$ level of significance level to determine whether the company’s claim is valid. Given that the value of $m$ is $4.5$, find $p$-value of this test.</div><p style="text-align: right;">[4]</p></li></ol>

A gaming company claims that gamers spend an average of $m$ hours daily on a new game that they have launched. A random sample of $30$ gamers was surveyed and the time, $x$ hours, spent daily on the game was recorded. The results are summarized by

$\sum{x}=126$ , $\sum{{{x}^{2}}}=550.1525$.

A test is carried out, at the $5\%$ significance level, to determine whether the average time spent daily on the game has been overstated by the company.

A second sample of $50$ gamers is surveyed and the time, $y$ hours, spent daily on the game was recorded, and the results are summarized by

Calculating and Interpreting $p$-Values

Estimating Parameters from Summarised and Coded Data

One-Tailed Hypothesis Tests

Unbiased Estimates of Population Parameters

Parameter Bounds from Hypothesis Tests

Combining Samples for Parameter Estimation

Hypothesis Tests Using Critical Regions and $p$-Values

Null and Alternative Hypotheses

Two-Tailed Hypothesis Tests

Large-Sample Mean Tests with Estimated Variance

2008 MJC MYE P2 Q11

Hypothesis Testing

Sampling

<ol class="abc-list"><li><div class="list-item-paragraph">Experimental data concerning a variable $X$, which measures the reliability of a certain electronic component, is as follows: $\sum{x}=1164.2$, $\sum{{{x}^{2}}}=13911.6$, $n=100$. Calculate the unbiased estimates of the population mean and variance from these figures. Explain whether, on the evidence of this sample, you would reject the hypothesis that the mean value of $X$ is $12$ at $5\%$ level of significance.</div></li><li><div class="list-item-paragraph">A second random sample of $50$ electronic components of this type is to be tested and the measurement, $y$, are summarized by</div><div class="list-item-paragraph" style="text-align: center">$\sum{y}=582.1$, $\sum{{{y}^{2}}}=7055.9$.</div><div class="list-item-paragraph">Combining the two samples into a single sample, carry out a test, at the $5\%$ significance level, of the same null and alternative hypotheses.</div></li><li><div class="list-item-paragraph">Figures collected over a long period have established that the mean and standard deviation of $X$ are $12$ and $2$ respectively. After a change in the manufacturing process, it is expected that the mean will have increased, but it may be assumed that the standard deviation remains equal to $2$. A sample of $n$ values of $X$ is taken, with sample mean $m$; if $m$ is greater than some critical value, it will be accepted that the mean has in fact increased, but if $m$ is less than the critical value, the increase is not established. State carefully appropriate null and alternative hypotheses for this situation, and find, in terms of $n$, the critical value for a $1\%$ significance level.</div></li></ol>

Critical Values and Critical Regions

NYJC Hypothesis Testing Tutorial Q8

<p>A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of $150$ hours. A random sample of $70$ batteries of this type is tested and the lifetime, $x$ hours, of each battery is measured. The results are summarized by</p>
<p style="text-align: center">$\sum{x}=10317$ , $\sum{{{x}^{2}}}=1540231$.</p>
<p>The population mean lifetime is denoted by $\mu $ hours. The null hypothesis $\mu =150$ is to be tested against the alternative hypothesis $\mu <150$. Find the $p$-value of the test and state the meaning of this $p$-value in the context of the question.</p>
<p>A second random sample of $50$ batteries of this type is tested and the lifetime, $y$ hours, of each battery is measured, with results summarized by</p>
<p style="text-align: center">$\sum{y}=7331$ , $\sum{{{y}^{2}}}=1100565$.</p>
<p>Combining the two samples into a single sample, carry out a test, at the $10\%$ significance level, of the same null and alternative hypotheses.</p>

A consumer association is testing the lifetime of a particular type of battery that is claimed to have a lifetime of $150$ hours. A random sample of $70$ batteries of this type is tested and the lifetime, $x$ hours, of each battery is measured. The results are summarized by

$\sum{x}=10317$ , $\sum{{{x}^{2}}}=1540231$.

The population mean lifetime is denoted by $\mu $ hours. The null hypothesis $\mu =150$ is to be tested against the alternative hypothesis $\mu <150$. Find the $p$-value of the test and state the meaning of this $p$-value in the context of the question.

A second random sample of $50$ batteries of this type is tested and the lifetime, $y$ hours, of each battery is measured, with results summarized by

$\sum{y}=7331$ , $\sum{{{y}^{2}}}=1100565$.

Combining the two samples into a single sample, carry out a test, at the $10\%$ significance level, of the same null and alternative hypotheses.

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