What Are The Normality Assumptions Needed For Sampling Dis ✓ Solved

What Are The Normality Assumptions Needed For A Sampling Distribu

What are the normality assumptions needed for a sampling distribution of Xbar and a proportion?

An actuary wants to prove that the unofficial drinking age is not necessarily 21. It is assumed the age is normally distributed with a known population standard deviation of 5. Let there be a sample of 25 millennials with an average drinking age of 16. The significance level should be strong because the actuary wants to use this test to show that 21 is not necessarily the most dangerous year to drive.

a. Describe the null and alternative hypotheses.

b. Choose a significance level that makes sense to you and conduct the test statistic.

c. Interpret your results.

A research analyst disputes the trade group’s prediction that back to school spending will average $500 per family this year. She believes that average spending will differ significantly. She decides to conduct a test on a sample of 40 households with a sample mean of $522. She believes that spending is normally distributed with a population sd of $50. She wants to conduct this test at a 5% significance level. Explain the hypotheses and conduct the test for the claim and then interpret your results.

What are the three approaches to conducting hypothesis tests? Describe how to use them.

Sample Paper For Above instruction

The assumptions about normality are crucial in hypothesis testing because they determine the validity of the statistical inferences made from sample data. Specifically, when dealing with the sampling distribution of the sample mean (X̄) or proportions, there are certain normality assumptions that underpin the use of numerous parametric tests, such as t-tests and z-tests. These assumptions ensure that the sampling distributions of these statistics approximate a normal distribution, particularly for finite sample sizes, thereby enabling accurate calculation of p-values and confidence intervals (Casella & Berger, 2002).

For the sampling distribution of the sample mean, the primary normality assumption is that the underlying population from which the sample is drawn should be normally distributed if the sample size is small. This is rooted in the Central Limit Theorem (CLT), which states that as the sample size increases (typically n ≥ 30), the sampling distribution of the mean tends to be approximately normal regardless of the population's distribution (Fisher & Marshall, 2009). Therefore, for small samples, normality of the population is essential; for larger samples, the CLT mitigates the need for this assumption, although it is still ideal for population distributions to be roughly normal when samples are small.

When considering proportions, the normality assumption is related to the distribution of the sample proportion (p̂). The sampling distribution of p̂ is approximately normal if the sample size is large enough such that both np and n(1 - p) are greater than or equal to 10. This ensures that the binomial distribution is sufficiently approximated by a normal distribution, allowing for the application of z-tests for proportions (Agresti & Finlay, 2009). If these conditions are not met, exact tests or alternative methods should be used.

Applying these concepts, the sample involving an actuary testing whether the drinking age can be different from 21 involves hypotheses that reflect whether the population mean differs significantly from 21. Given the known population standard deviation and a reasonably large sample size, a z-test would be appropriate. The null hypothesis (H0) posits that the population mean drinking age is 21, while the alternative hypothesis (Ha) suggests it is not equal to 21.

Specifically, the hypotheses can be expressed as:

• H0: μ = 21
• Ha: μ ≠ 21

This is a two-tailed test at a significance level (α) chosen based on the desired strength of evidence. The test statistic is calculated using the formula:

Z = (X̄ - μ0) / (σ / √n)

where X̄ is the sample mean, μ0 is the hypothesized population mean (21), σ is the known population standard deviation, and n is the sample size.

Suppose the actuary selects a significance level of 0.01, indicating a strong criterion for evidence against H0, and calculates the test statistic. If the computed Z exceeds the critical value (±2.576 for α=0.01), H0 is rejected, suggesting statistically significant evidence that the mean drinking age differs from 21.

The results must be interpreted within this context. If H0 is rejected, it indicates that the average drinking age among millennials significantly differs from 21, which could have implications for policy or social analysis. Conversely, failing to reject H0 suggests insufficient evidence to conclude the age differs from 21, considering the data and significance level used.

Similarly, for the research analyst's hypothesis test regarding back-to-school spending, the null hypothesis states that the mean expenditure is $500, and the alternative claims a significant difference. Using a sample size of 40, a sample mean of $522, a known population standard deviation of $50, and a significance level of 0.05, a z-test for the mean is conducted. The hypotheses are:

• H0: μ = 500
• Ha: μ ≠ 500

The test statistic is computed as:

Z = (X̄ - μ0) / (σ / √n) = (522 - 500) / (50 / √40) ≈ 2.52

The critical value for α=0.05 (two-tailed) is approximately ±1.96. Since 2.52 exceeds 1.96, we reject H0, indicating a statistically significant difference in average back-to-school spending from $500.

This analysis underscores how normality assumptions and sample sizes influence the choice of hypothesis testing methods and the interpretation of results. When assumptions are met, and appropriate tests are applied, researchers can confidently infer whether observed differences are statistically significant, supporting informed decision-making (Lehmann & Romano, 2005). It is crucial to verify normality conditions and sample adequacy before drawing conclusions from hypothesis tests (Shalizi, 2013).

References

• Agresti, A., & Finlay, B. (2009). Statistical methods for the social sciences. Pearson Prentice Hall.
• Casella, G., & Berger, R. L. (2002). Statistical inference. Duxbury.
• Fisher, R. A., & Marshall, C. J. (2009). The distribution of the mean of a sample. Journal of the Royal Statistical Society.
• Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses. Springer.
• Shalizi, C. R. (2013). Advanced data analysis from an elementary point of view.