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Lucky numbers If people choose lottery numbers at random, the last digit should be equally likely to be any of the ten digits from 0 to \(9 .\) Let \(p\) measure the proportion of choices that end with the digit 7\. If choices are random, we would expect \(p=0.10\), but if people have a special preference for numbers ending in 7 the proportion will be greater than 0.10 . Suppose that we test this by asking a random sample of 20 people to give a three-digit lottery number and find that four of the numbers have 7 as the last digit. Figure 5.13 shows a randomization distribution of proportions for 5000 simulated samples under the null hypothesis \(H_{0}: p=0.10\).(a) Use the sample proportion \(\hat{p}=0.20\) and a stan- (c) Compare the p-value obtained from the nordard error estimated from the randomization \(\quad\) mal distribution in part (b) to the p-value shown distribution to compute a standardized test \(\quad\) for the randomization distribution. Explain why statistic. \(\quad\) there might be a discrepancy between these two (b) Use the normal distribution to find a p-value for values. an upper tail alternative based on the test statistic found in part (a).

Step by step solution

01

Calculate Sample Proportion

Given that four out of twenty people chosen at random have the number 7 as their last digit, we can calculate the sample proportion \(\hat{p}\) using the formula \(\hat{p}= \frac{x}{n}\) where \(x\) is the successful outcomes and \(n\) is the total outcomes. Here, \(x = 4\) and \(n = 20\). So, \(\hat{p}= \frac{4}{20} = 0.20\).

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Step 2. Calculate Standardized Test Statistic

To calculate the standardized test statistic we use the formula \(\frac{\hat{p} - p_{0}}{SE_{p_{0}}}\) where \(\hat{p}\) is the sample proportion, \(p_{0}\) is the expected proportion and \(SE_{p_{0}}\) is the standard error estimated from the randomization. Given that \(p_{0} = 0.10 \) and \(SE_{p_{0}}\) would be \(\sqrt{p_{0}(1 - p_{0}) / n}\), we plug this values in and calculate the test statistic.

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Step 3. Calculate p-value from a Normal Distribution

The p-value is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. With a normal distribution, this value is obtained by checking the statistical tables or using calculators that are programmed to find p-values. The test statistic, which was derived in step 2, is used in finding this p-value.

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Step 4. Compare the p-value with the Randomization Distribution

The randomization distribution is a hypothetical distribution of sample statistics that would be seen if the null hypothesis were true. The p-value derived from the normal distribution is compared to the p-value shown for the randomization distribution. Discrepancies between the two might occur due to the deviation from normality or differences in sample sizes.

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Key Concepts

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

Randomization Distribution

The randomization distribution is the foundation of hypothesis testing in statistics. Imagine you have a particular hypothesis, say, people pick lottery numbers randomly. To test this hypothesis, you simulate your experiment many times under the null hypothesis, which assumes no effect or no difference. In our example, this would mean a preference for any last digit is equally likely.
When you generate a randomization distribution, you create thousands of possible outcomes by simulating samples under the assumption that the null hypothesis is true.
This helps in visualizing what the distribution of sample proportions looks like if there were no special preference for the number seven.

• This distribution provides a picture of sampling variability if the selection of last digits were truly random.
• By comparing the actual sample results with this distribution, we can see how likely or unlikely our observed results are.

It's like taking a look at what we'd get if we ran the test over and over again under the no-preference assumption.

Sample Proportion

The sample proportion, represented by \(\hat{p}\), is the fraction of the sample that has the characteristic of interest. It's a way to estimate the population proportion by using a smaller set of data. This makes the concept very practical for real-world applications.
In the lottery number example, the sample proportion is the proportion of people who chose a number ending in 7 out of the 20 sampled individuals.
It is defined mathematically as:
\[\hat{p} = \frac{x}{n}\]Where:

• x is the number of "successful" outcomes, or the number of people who picked a digit ending in 7.
• n is the total number of people sampled.

For our case, x equals 4, and n is 20, so \(\hat{p} = 0.20\).
This sample proportion helps determine if the preference for choosing the number 7 is significantly different from random choice expectations.

Standardized Test Statistic

Standardized test statistics are used to determine how far away a sample statistic is from the hypothesized population parameter, in terms of standard errors. It's a crucial component in hypothesis testing as it helps you determine if your observed data differ significantly from what the null hypothesis predicts.
The formula to find the standardized test statistic (Z) in the context of sample proportions is:
\[Z = \frac{\hat{p} - p_0}{SE_{p_0}}\]Where:

• \(\hat{p}\) is the sample proportion, representing observed data.
• \(p_0\) is the hypothesized population proportion under the null hypothesis.
• \(SE_{p_0}\) is the standard error of the proportion, calculated as \(SE_{p_0} = \sqrt{\frac{p_0 (1-p_0)}{n}}\).

For our exercise, with \(\hat{p} = 0.20\), \(p_0 = 0.10\), and \(n = 20\), you calculate the standard error and then the Z test statistic.
A high absolute value of Z indicates that the proportion from the sample differs significantly from the expected proportion, helping you make conclusions about lottery number preferences.