Question

A researcher estimated the height of Tamarack trees on Beaver Island (Michigan, USA) along two different sites. Six trees were sampled randomly at each of the two sites. Below is the diameter of each tree (cm)

48.1, 56.8, 48.0, 53.4, 57.8, 56.3, 62.4, 50.8, 60.1, 57.2, 55.4, 50.1

a. State the hypotheses being tested.

b.) Calculate the appropriate statistical test and degrees of freedom (assume the data were not normally distributed).

c). Draw inferences from your test and state your conclusions at an alpha of 0.05.

Answer 1

Solution:-

a)

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

Null hypothesis: u₁ = u ₂
Alternative hypothesis: u₁ u ₂

Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.

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.

b)

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

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 2.6957
DF = 10
t = [ (x₁ - x₂) - d ] / SE

t = - 0.975

where s₁ is the standard deviation of sample 1, s₂ is the standard deviation of sample 2, n₁ is the size 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 the population means, and SE is the standard error.

Since we have a two-tailed test, the P-value is the probability that a t statistic having 10 degrees of freedom is more extreme than -0.975; that is, less than -0.975 or greater than 0.975.

Thus, the P-value = 0.353

Interpret results. Since the P-value (0.353) is greater than the significance level (0.05), we have to accept the null hypothesis.

c) From the above test we have sufficient evidence in the favor of the claim that means at two different sites are equal.