According To The Past 5 Years Of Experience, A Professor Kno

According To The Past 5 Years Of Experience A Professor Knows That

According to the past 5 years of experience, a professor knows that the average hours his students spend on the final project is 15, with a standard error of the mean of 0.9. The professor sampled 50 students this semester and found an average of 14 hours spent on the project. The task involves hypothesis testing to determine if the students are spending less time on the final project this semester.

First, the professor needs to formulate the null and alternative hypotheses. The null hypothesis (H₀) assumes that the average time students spend on the final project remains unchanged from past years, i.e., 15 hours. The alternative hypothesis (H₁) suggests that the average time has decreased this semester.

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The appropriate null hypothesis (H₀) for this scenario is that the mean hours spent by students on the final project is equal to 15 hours, expressed as:

  • H₀: μ = 15

The alternative hypothesis (H₁) posits that the mean hours have decreased, expressed as:

  • H₁: μ

Using a significance level of p = 0.05, we then find the critical value corresponding to a one-tailed test, as the hypothesis test is directional (checking if the mean has decreased). For a standard normal distribution, the critical z-value at p = 0.05 is approximately -1.645.

Next, we calculate the z-statistic to determine how many standard errors the sample mean (14 hours) is away from the population mean (15 hours). The formula for the z-statistic in this case is:

z = (x̄ - μ₀) / (s.e.)

where:

  • x̄ = 14 (sample mean)
  • μ₀ = 15 (population mean under null hypothesis)
  • s.e. = 0.9 (standard error of the mean)

Substituting these values, we find:

z = (14 - 15) / 0.9 = -1 / 0.9 ≈ -1.11

Comparing the calculated z-value (-1.11) to the critical z-value (-1.645), we observe that -1.11 is greater than -1.645; thus, it does not fall into the critical region. In hypothesis testing, this means we fail to reject the null hypothesis at the 0.05 significance level.

Therefore, based on this analysis, there is insufficient evidence to conclude that students are spending less time on the final project this semester. The observed decrease from an average of 15 hours to 14 hours could be attributed to random variation within the sampling error.

In conclusion, while the sample suggests a potential reduction in time spent, the statistical test indicates that the difference is not statistically significant at the 5% level. The students' average effort remains consistent with past years, and further analysis with larger samples or different methodologies could provide additional insights.

References

  • Gibbons, J. D., & Chakraborti, S. (2011). Nonparametric Statistical Inference (5th ed.). Chapman and Hall/CRC.
  • Laerd Statistics. (2018). One Sample Z-test. Retrieved from https://statistics.laerd.com/statistical-guides/z-test-one-sample.php
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). SAGE Publications.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • Hogg, R. V., McKean, J., & Craig, A. T. (2013). Introduction to Mathematical Statistics (7th ed.). Pearson.
  • Triola, M. F. (2018). Elementary Statistics (13th ed.). Pearson.
  • Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Methods and Data Analysis (6th ed.). Brooks/Cole.
  • Everitt, B. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.

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