A Recent National Survey Found That High School Students Wat

A recent national survey found that high school students watched an Av

A recent national survey found that high school students watched an average of 7.8 DVDs per month with a population standard deviation of 1.00. A random sample of 30 college students revealed that the mean number of DVDs watched last month was 7.30. At the 0.05 significance level, we want to determine if college students watch fewer DVDs per month than high school students.

To address this, we will perform a hypothesis test for the population mean, assuming the population standard deviation is known. The hypotheses are:

  • Null hypothesis, H0: μ = 7.8 (college students watch the same number or more DVDs than high school students)
  • Alternative hypothesis, Ha: μ

Given data:

  • Population mean (high school students), μ0 = 7.8
  • Sample mean, x̄ = 7.30
  • Sample size, n = 30
  • Population standard deviation, σ = 1.00

Calculation of the Test Statistic

The z-test statistic is calculated using the formula:

z = (x̄ - μ0) / (σ / √n)

Plugging in the values:

z = (7.30 - 7.8) / (1.00 / √30)

z = (-0.50) / (1.00 / 5.4772)

z = (-0.50) / 0.1826

z ≈ -2.74

The value of the test statistic is approximately -2.74.

Determining the p-value

Since the alternative hypothesis is that the mean is less than 7.8, this is a left-tailed test. Using standard normal distribution tables or statistical software, we find the p-value corresponding to z = -2.74.

Consulting a standard normal distribution table or calculator, p ≈ 0.0030.

This p-value indicates a very low probability of observing such a sample mean if the true population mean were 7.8, supporting the alternative hypothesis.

Conclusion

At the 0.05 significance level, the p-value (≈ 0.003) is less than 0.05. Therefore, we reject the null hypothesis and conclude that there is statistically significant evidence to suggest that college students watch fewer DVDs per month than high school students.

Additional Explanation of the Heckscher-Ohlin Theorem and Cross-Hauling

The Heckscher-Ohlin theorem explains patterns of international trade based on factor endowments. It predicts that countries will specialize in and export goods that intensively use their abundant factors of production and import goods that require factors they are relatively scarce in. This theorem implies that the pattern of trade reflects comparative advantages rooted in factor endowments rather than differences in technology or preferences alone.

Regarding cross-hauling, which refers to the simultaneous import and export of similar goods between countries, the Heckscher-Ohlin theorem suggests that such phenomena should not be prevalent under certain conditions. When trade is driven primarily by factor endowments, countries tend to specialize and produce distinct goods aligned with their comparative advantages. Cross-hauling occurs when both countries trade similar goods, which can happen if factors of production are similar or if there are product differentiations and consumer preferences that deviate from what basic factor endowment models predict.

Therefore, according to the Heckscher-Ohlin theorem, cross-hauling should be less common because trade patterns should reflect clear specialization based on factor endowments, and extensive cross-trading of similar goods would indicate other underlying factors such as product differentiation, economies of scale, or specific preferences, rather than pure factor endowment advantages.

References

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  • Helpman, E., & Krugman, P. R. (1985). Market Structure and Foreign Trade: Increasing Returns, Imperfect Competition, and the International Economy. Harvard University Press.
  • Samuelson, P. A. (1948). International Trade and the Equalisation of Factor Prices. The Economic Journal, 58(230), 163-184.
  • PLS Data Analysis. (2022). Conducting hypothesis tests for means. PLS Data Analysis Guides.
  • Frankel, J. A. (1997). Regional Trading Blocs in the World Economic System. The Brookings Institution.
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  • Jones, R. W. (2000). The Welfare Economics of International Trade. Journal of International Economics, 50(1), 3-24.

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