Question

Using either hypothesis testing or confidence interval method solve: The average size of a farm green county is 199 acres, and the average size of a farm in yellow county is 191. Assume the data were obtained from two samples with standard deviations of 12 acres and 38, respectively, and the sample sizes are 10 farms from Green county and 8 farms from in Yellow County. Can it be concluded at alpha= 0.05 that the average size of the farms in the two counties are different. Assume the populations are normally distributed.

Answer 1

Solution:-

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 0.10. 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).

SE = sqrt[(s₁²/n₁) + (s₂²/n₂)]
SE = 13.9607
DF = 16
t = [ (x₁ - x₂) - d ] / SE

t = 0.573

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 16 degrees of freedom is more extreme than -0.573; that is, less than -0.573 or greater than 0.573.

Thus, the P-value = 0.575.

Interpret results. Since the P-value (0.575) is greater than the significance level (0.05), we have to accept the null hypothesis.

From the above test we do not have sufficient evidence in the favor of the claim that the average size of the farms in the two counties are different.