Question

6.222 Metal Tags on Penguins and Length of Foraging Trips Data were collected over a 10-year timespan from a sample of 100 penguins that were randomly given either metal or electronic tags. One variable examined is the length of foraging trips. Longer foraging trips can jeopardize both breeding success and survival of chicks waiting for food. Mean length of 344 foraging trips was 12.70 days for metal-tagged penguins. Mean length of 512 foraging trips was 11.60 days for electronic-tagged penguins. An estimate of the standard error for this difference of means is SE 0.283. 1. a. We will address the question of whether foraging trips are longer on average among metal-tagged penguins than among electronic-tagged penguins. State hypotheses in terms of two means. b. Calculate the sample statistic - Calculate a t-test statistic, using the given estimate of SE. c. d. What are the correct (conservative) degrees of freedom for a t-distribution for this test? Use t-distribution methods to find the p-value and draw a rough curve with appropriate shaded region. You may give either a precise p-value from software or bounds on a p-value from Table A. e. f. State the conclusion of the test in context, using nontechnical language.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: u_Metal< u_Electronic
Alternative hypothesis: u_Metal > u_Electronic

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.

Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).

b) The sample statistics $\bar{x}_{M}-\bar{x}_{E}$ is 1.10.

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 0.283

c)

t = [ (x₁ - x₂) - d ] / SE

t = 3.89

d)

DF = 854

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is thesize of sample 1, n₂ is the size of sample 2, x₁ is the mean of sample 1, x₂ is the mean of sample 2, d is the hypothesized difference between population means, and SE is the standard error.

The observed difference in sample means produced a t statistic of 3.89.

e) Therefore, the P-value in this analysis is less than 0.0001.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

f) From the above test we have sufficient evidence in the favor of the claim that foraging trips are longer on average among metal tagged penguins than among electronic-tagged penguins.