Suppose That The Average Weekly Wage For Women In The US Is

Suppose That The Average Wekly Wage For Women In the Us Is Around 740

Suppose that the average weekly wage for women in the US is around $740 (see BLS report) while the average weekly wage for men is around $900. Suppose that the respective sample sizes are 1000 each and the standard deviations are $300 and $450. To test for the equality of the wages for the two populations, what is the test statistic? Answer to two decimal places. Hint: your number should be negative.

Suppose you have a product which you sell in two markets of comparable size and you want to test the hypothesis that the demand for the product is the same in the two markets. In practice, you test that the quantity demanded is the same and find that for a sample of 17 different time periods with different prices, the average quantity demanded in the first market is 1765 and in the second market it is 1688. The standard deviation of the differences is 60.5. What is the standard error? Answer to two decimal places.

Suppose you have a product which you sell in two markets of comparable size and you want to test the hypothesis that the demand for the product is the same in the two markets. In practice, you test that the quantity demanded is the same and find that for a sample of 17 different time periods with different prices, the average quantity demanded in the first market is 1765 and in the second market it is 1688. The standard deviation of the differences is 60.5. True or False: we reject the null hypothesis of equal means for quantity demanded at the .05 level of significance. True or False.

What are examples of variables that follow a binomial probability distribution? What are examples of variables that follow a Poisson distribution? When might you use a geometric probability?

Paper For Above instruction

The analysis of statistical hypotheses and probability distributions is fundamental in decision-making processes across various fields such as economics, business, and social sciences. This paper addresses several statistical issues: calculating the test statistic for difference in population means, determining the standard error for paired sample differences, evaluating the rejection of a null hypothesis at a specified significance level, and identifying appropriate probability distributions for different types of variables. Each topic is discussed with detailed explanation and relevant formulas, supported by scholarly references.

Comparison of Population Wages: Test Statistic Calculation

The first problem involves comparing the average weekly wages of women and men in the US to assess whether these wages are statistically significantly different. Given that the sample sizes are both 1,000, the sample means are $740 for women and $900 for men, and the standard deviations are $300 and $450, respectively, the appropriate test is a two-sample z-test for equal means. The test statistic is calculated as:

z = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Substituting the values:

z = (740 - 900) / √((300)²/1000 + (450)²/1000) = (-160) / √(90 + 202.5) = (-160) / √292.5 ≈ (-160) / 17.10 ≈ -9.36

This indicates a large magnitude for the test statistic, and since it is negative (as expected, given the alternative hypothesis that wages are not equal), it strongly suggests rejecting the null hypothesis of equal wages at typical significance levels.

Standard Error for Paired Differences

The second problem pertains to the calculation of the standard error (SE) for the average difference in demands in two markets. Given the mean demands of 1765 and 1688, a sample size of 17, and a standard deviation of differences of 60.5, the standard error is computed as:

SE = s / √n

where s is the standard deviation of differences:

SE = 60.5 / √17 ≈ 60.5 / 4.123 ≈ 14.68

Therefore, the standard error of the mean difference is approximately 14.68, providing a measure of variability in the estimated difference across the time periods.

Hypothesis Testing at the 0.05 Significance Level

Using the calculated mean difference of 77 (1765 - 1688) and the standard error of approximately 14.68, we conduct a t-test to assess whether the observed difference is statistically significant at the 0.05 level. The t-statistic is:

t = (x̄₁ - x̄₂) / SE ≈ 77 / 14.68 ≈ 5.25

With 16 degrees of freedom (n - 1 = 17 - 1), the critical t-value at the 0.05 significance level (two-tailed) is approximately 2.12. Since 5.25 > 2.12, we reject the null hypothesis of equal means. Therefore, the statement "we reject the null hypothesis at the .05 level of significance" is True.

Variables Following Specific Probability Distributions

In probability theory, certain types of variables are modeled using specific distributions based on the nature of the data. Variables that follow a binomial distribution are discrete variables representing the number of successes in a fixed number of independent Bernoulli trials with the same probability of success. Examples include:

  • Number of defective items in a batch
  • Number of heads in a series of coin flips
  • Number of correct answers on a multiple-choice test

Variables that follow a Poisson distribution are discrete variables representing the number of occurrences of an event in a fixed interval or space when events occur randomly and independently at a constant average rate. Examples include:

  • The number of emails received in an hour
  • The number of decay events in a radioactive sample
  • The count of cars passing a tollbooth in a minute

Geometric probability models the number of trials needed to achieve the first success in a sequence of independent Bernoulli trials with the same probability of success. It is used when the interest is in the trial count until the first success occurs, e.g.,

  • The number of coin flips until the first head
  • The number of sales calls until a customer agrees to purchase

These distributions are essential tools in modeling and analyzing various stochastic processes depending on the context and data characteristics.

Conclusion

Statistical testing of population parameters allows researchers and analysts to draw meaningful inferences about real-world phenomena. Accurate computation of test statistics, standard errors, and understanding appropriate probability distributions are crucial for valid conclusions. As demonstrated, the t-test and z-test facilitate hypothesis testing for population means, while distributions such as binomial, Poisson, and geometric are suited to different types of discrete variables and scenarios, enhancing decision-making processes based on data analysis.

References

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  • Libai, B., Muller, E., & Peres, R. (2009). The Role of Word of Mouth in Explosive Diffusion. Journal of Marketing, 73(5), 1-15.
  • Mendenhall, W., Ott, L., & Anderson, R. (2006). Elementary Statistics: A Step By Step Approach. Brooks/Cole.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
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  • Devore, J. L. (2011). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Ross, S. (2014). Introduction to Probability Models. Academic Press.

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