Sample Of 47 Adults Tested, 9 Of Them Are Found Active

A Sample Of 47 Adults Have Been Tested And 9 Of Them Are Found To Have

A sample of 47 adults have been tested and 9 of them are found to have influenza. Find the P-Value, for hypothesis test to determine the proportion has been exposed to influenza differs from 8% .0024 .0048 .0262 .0524 Consider the Sum of Two Dice scenario below. Perform the required experiment and hypothesis test by completing the eight steps. Submit your document in Word or Rich text format (.rtf) as Lastname_M10800_InferenceProject. The calculations should be incorporated into the document or attached as an appendix in jpeg or pdf format. Sum of Two Dice Are your dice loaded? When playing with honest, or balanced, dice, the random variable X representing the sum of the numbers on the faces when two dice are rolled possesses the following probability distribution: x p(x) 0...........028 Furthermore, the mean of the distribution, μ = 7 and the standard deviation, σ = 2.4152. Examine the average sum of two dice rolled 40 times. Do your dice appear to be fair? Perform a test of the null hypothesis that the mean sum is 7. Steps for your analysis: 1. Roll two dice 40 times. Record the sum of numbers on the faces of the two dice each time. Find the mean sum for your sample. 2. Indicate the null and alternative hypotheses. 3. Indicate the distribution that the sampling distribution of your sample mean follows and justify your reasoning. In other words, will your test statistic follow a z distribution, a t distribution, or some other distribution, and how do you know? 4. Calculate your test statistic. 5. State the rejection region. 6. Compare the test statistic against the critical value for the rejection region, and state your conclusion. 7. Interpret your conclusion within the context of the experiment. 8. Determine the observed significance level (p-value) of your test.

Paper For Above instruction

Hypothesis testing is an essential aspect of statistical analysis, allowing researchers to make inferences about population parameters based on sample data. This paper addresses two related hypothesis tests: one concerning the proportion of adults exposed to influenza and the other assessing whether a pair of dice is loaded or biased. These tests not only demonstrate fundamental statistical concepts but also highlight their practical applications in public health and gaming fairness.

Part 1: Testing the Proportion of Influenza Exposure

The first scenario involves a sample of 47 adults, with 9 found to have influenza. The objective is to determine whether the true proportion of adults exposed to influenza differs from 8%. To achieve this, a hypothesis test for a population proportion is conducted.

The null hypothesis (H₀) posts that the proportion of adults with influenza is 8% (p = 0.08), while the alternative hypothesis (H₁) suggests that this proportion is different from 8% (p ≠ 0.08).

Using the sample data, the sample proportion (p̂) is calculated as 9/47 ≈ 0.1915. The standard error (SE) for the proportion under H₀ is computed as:

SE = √[p₀(1 - p₀)/n] = √[0.08 * 0.92 / 47] ≈ 0.0393.

The z-test statistic is then calculated as:

z = (p̂ - p₀) / SE = (0.1915 - 0.08) / 0.0393 ≈ 3.10.

Referring to standard normal distribution tables, the p-value for a two-tailed test with z ≈ 3.10 is approximately 0.002, which matches the provided options (0.0024, 0.0048, 0.0262, 0.0524). Since 0.002 is less than the typical significance level of 0.05, we reject H₀, indicating that the proportion of adults exposed to influenza significantly differs from 8%.

Part 2: Assessing Dice Fairness Through Mean Sum Testing

The second scenario involves testing whether a pair of dice are balanced based on the average sum of 40 rolls. The known population mean (μ) for a fair pair of dice is 7, with a standard deviation (σ) of approximately 2.4152. The null hypothesis (H₀) states that the true mean sum is 7, while the alternative hypothesis (H₁) is that it differs from 7.

Following the experimental step, you roll two dice 40 times and record the sums. Suppose the sample mean sum calculated from these rolls is denoted as x̄.

The sampling distribution of the sample mean follows a normal distribution because of the Central Limit Theorem, which applies when the sample size exceeds 30, regardless of the population distribution. Since the population standard deviation (σ) is known, the z-test is appropriate.

The z-test statistic is calculated as:

z = (x̄ - μ) / (σ / √n) = (x̄ - 7) / (2.4152 / √40).

Depending on your observed sample mean, this statistic is compared against critical z-values at significance levels such as 0.05 or 0.01. The rejection region for a two-tailed test at α = 0.05 is z 1.96.

If the calculated z falls beyond these bounds, we reject H₀, indicating evidence that the dice may be biased. Otherwise, we fail to reject H₀, supporting the fairness of the dice.

For example, if the observed mean sum is 7.5, then:

z = (7.5 - 7) / (2.4152 / √40) ≈ 1.31, which does not exceed 1.96, leading to a failure to reject the null hypothesis.

Similarly, calculating the p-value associated with the observed z allows for a more precise significance level interpretation.

Conclusion

In the influenza study, we found compelling evidence that the proportion exposed to influenza is significantly different from 8%, which has implications for public health policies. Conversely, in analyzing dice fairness, the results depend on the observed mean sum; based on standard thresholds, the tests can confirm or refute the fairness of the dice.

Both statistical tests exemplify the importance of hypothesis testing in analyzing real-world data, providing critical insights for decision-making in health and gaming settings.

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