"Time In Purgatory: Examining the Grant Lag for U.S. Patent Applications," published online by the Berkeley Electronic Press in 2003, examined grant lag (period of time between an initial patent application and its final granting, in months) for all patents on record between 1976 and 1996. There were well over 1 million patents during this period, and their grant lag time had mean 28.4 months, standard deviation 19.7 months.
a. For the 1,278 patents whose assignees (those who hold the property rights bestowed by the invention) were nonprofit organizations, the mean grant lag was 30.3 months. State the null and alternative hypotheses to see if the mean for nonprofit organizations differs significantly from the overall mean lag, 28.4 months. Report your z statistic.
b. What can you say about the size of the P-value for the test in part (a)?
c. What do you conclude about the mean grant lag time for nonprofit organizations?
d. For the 17,209 patents whose assignees were listed as "independent," the mean grant lag was 28.6 months. State the null and alternative hypotheses to see if the mean for independent organizations differs significantly from the overall mean lag, 28.4 months. Report your z statistic.
e. What can you say about the size of the P-value for the test in part (d)?
f. What do you conclude about the mean grant lag time for independent organizations?
g. Sample mean grant lag for patents whose assignees were from the government was further from the population mean than sample mean lag for nonprofit organizations. If the sample size was the same, would the P-value for the government be larger or smaller than the one for nonprofits?
h. The sample size for patents whose assignees were from the government was larger than the sample size for nonprofit organizations. If the sample mean was the same, would the P-value for the government be larger or smaller than the one for nonprofits?
i. Patents whose assignees were private firms made up 78% of all the patents recorded. Explain why we should not use a z procedure to test if their mean grant lag time (28.6 months) differs significantly from the overall mean lag, 28.4 months.
j. The authors present a histogram of grant lags, with the following caution: "The actual tail of the distribution continues beyond what is shown in the graph, as our data set includes patents with lags up to 1143 months long."11 Is the distribution left-skewed or right-skewed?
k. Would median grant lag be greater than or less than 28.4? Explain.