91Ó°ÊÓ

In Las Vegas and Atlantic City, New Jersey, tests are performed often on the various gaming devices used in casinos. For example, dice are often tested to determine if they are balanced. Suppose you are assigned the task of testing a die, using a two-tailed test to make sure that the probability of a 2 -spot is \(1 / 6 .\) Using the \(5 \%\) significance level, determine how many 2 -spots you would have to obtain to reject the null hypothesis when your sample size is \(\begin{array}{lll}\text { a. } 120 & \text { b. } 1200 & \text { c. } 12,000\end{array}\) Calculate the value of \(\hat{p}\) for each of these three cases. What can you say about the relationship between (1) the difference between \(\hat{p}\) and \(1 / 6\) that is necessary to reject the null hypothesis and (2) the sample size as it gets larger?

Step by step solution

01

Calculate critical value

The first task is to calculate the critical value. For a two tailed test with a significance level of \(5\%\), the confidence level is \(95\%\). We deduce from the z-score table that the critical value (z*) at \(95\%\) confidence level is \( \pm 1.96\).

02

Calculate \(\hat{p }\) for each sample size

We then use the critical value and the standard deviation formula \(\sqrt {\frac { p(1 - p) }{ n}} \) to solve for \(\hat{p}\). For each sample size, we calculate the number of 2-spots (x) that corresponds to the critical value of a \(5\) percent signficance level. Hence, we form the following three proportional equations: for sample size \( n = 120, 1.96 = \frac {x - 120/6 }{ \sqrt {(120/6)*(1 - 1/6)/120} },\) for sample size \( n = 1200, 1.96 = \frac {x - 1200/6 }{ \sqrt {(1200/6)*(1 - 1/6)/1200} },\) and for sample size \( n = 12000, 1.96 = \frac {x - 12000/6 }{ \sqrt {(1200/6)*(1 - 1/6)/12000} }.\) We solve these equations for \(x\) and round to the next whole number because we cannot have a fraction of a dice roll. Divide each \(x\) by its corresponding \(n\) to find \(\hat{p}\).

03

Comment on the relationship

We observe the \(\hat{p }\) values obtained for different sample sizes. As the sample size increases, for the null hypothesis to be rejected, the difference between \(\hat{p }\) and \(1/6\) becomes smaller. This signifies that as we increase the sample size, our result becomes more precise; we need lesser deviation from the expected probability to reject the null hypothesis. This is consistent with the law of large numbers, which states that as the size of a sample becomes larger, the estimate of the sample mean will be more accurately reflect the population mean.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significance Level

In the realm of hypothesis testing, the significance level is a crucial concept. It defines the probability of rejecting the null hypothesis when it is true. Typically denoted as \( \alpha \), this value is often set at 0.05, or 5%. This means there's a 5% risk of concluding there is an effect when there is none.

In the context of our die-testing problem, the significance level determines the threshold for our test. If our calculated result falls beyond this threshold, we reject the null hypothesis, suggesting that the die may not be balanced.

• A significance level of 0.05 is commonly used in many experiments and indicates a 95% confidence in the results.
• This level helps in setting the stage for determining the critical value of a test.

Understanding the significance level helps us control the possible error in our conclusions.

Two-Tailed Test

A two-tailed test investigates both directions of an effect. We are concerned with deviations on either side of the theoretical probability.

In the die-testing scenario, a two-tailed test checks if the frequency of rolling a 2-spot is either significantly too high or too low compared to the expected probability of \( \frac{1}{6} \). This kind of test is particularly important when we have no inherent suspicion about the direction of the test statistic's deviation.

• It requires setting up the null hypothesis that the outcome is equal to what is expected (like \( p = \frac{1}{6} \) ).
• The alternative hypothesis would be that the probability is not equal to the expected rate.

Using a two-tailed test ensures that unusual deviations in both directions are captured, providing a more robust assessment.

Critical Value

The critical value is a point on the test distribution. It represents the boundary or threshold at which we decide whether to reject the null hypothesis. In our example, with a 5% significance level, the critical values correspond to \( \pm 1.96 \) when referring to the standard normal distribution.

These values are used to determine how far our sample mean can deviate from the expected mean before the results are considered statistically significant.

• The critical value varies based on the chosen significance level and the type of test (one-tailed vs. two-tailed).
• Finding the critical value involves consulting a z-table or a similar statistical tool depending on the distribution of your data.

Setting and understanding the critical value is crucial for interpreting your test's results appropriately.

Sample Size

Sample size is the number of observations in a sample, denoted as \( n \). It plays a significant role in hypothesis testing by impacting the test's precision. A larger sample size generally yields more reliable results.

In our die test, as we increase the sample sizes from 120 to 12,000, the difference needed between \( \hat{p} \) and \( \frac{1}{6} \) to reject the null hypothesis decreases. This aligns with the law of large numbers, stating that as a sample size grows, the estimate becomes more reliable.

• Larger sample sizes lead to smaller margins of error and more precise estimates.
• However, larger samples require more resources, whether in time, money, or effort.

Thus, deciding on a sample size is a balance between precision and available resources.