Question

Qi: Solve the modified exercise 8.120 (p. 422) in your textbook. (5 points) Shaft graves in ancient Greece. Archeologists have discovered a rise in shaft graves during the Middle Helladic period in ancient Greece (i.e., around 2000 BC). Shaft graves are named for the beautifully decorated sword shafts that are buried along with the bodies. An analysis of shaft graves was carried out and published in the American Journal of Archaeology (Jan. 2014). The table below gives the number of shafts buried at each of 13 recently discovered grave sites. 1 231 5 6241 24 2 9 Let σ represent the standard deviation of the number of sword shafts for the population of all shaft graves in ancient Greece. Is there sufficient evidence to conclude that the true standard deviation of the number of sword shafts is less than 3?

Answer 1

Null hypothesis H0: $\sigma = 3$

Alternative hypothesis H0: $\sigma < 3$

We shall choose $\alpha$ = 0.05

The test statistic to test for the standard deviation will follow a chi-square distribution with n-1 degree of freedom.

Degree of freedom = n - 1 = 13 - 1 = 12

Mean shafts buried = (1 + 2 + 3 + 1 + 5 + 6 + 2 + 4 + 1 + 2 + 4 + 2 + 9)/13 = 3.2308

Variance of shafts buried, s² = [(1 - 3.2308)² + (2 - 3.2308)² + (3 - 3.2308)² + (1 - 3.2308)² + (5 - 3.2308)² + (6 - 3.2308)² + (2 - 3.2308)² + (4 - 3.2308)² + (1 - 3.2308)² + ( 2 - 3.2308)² + (4 - 3.2308)² + (2 - 3.2308)² + (9 - 3.2308)² ]/12

= 5.5256

Test statistic, $\chi^2 = (n-1)s^2 / \sigma^2 = (13-1) * 5.5256 / 3^2 = 7.3675$

P-value = P($\chi^2 < 7.3675$) for df = 12 is  0.1676

Since, p-value is greater than the significance level of 0.o5, we fail to reject H0 and conclude that there is no significant evidence that $\sigma < 3$ or there is no significant evidence that true standard deviation of the number of sword shafts is less than 3 .