Question

You are a senior in the agricultural sciences department of NCSU doing an internship at a local dairy farm. Farmer Bill claims that the milk produced by his grass-fed jersey cattle has higher protein content than the milk made by his grain-fed jersey cattle. Based on your test statistic of 1.98, which of the following is true?

a. There is sufficient evidence at an α=0.05 level to suggest the population mean protein content of milk from grass-fed cattle is greater than the population mean protein content of milk from grain-fed cattle, based on the collected sample.
b. A 90% confidence interval for μ1-μ2 calculated based on the collected sample would contain 0.
c. The sample mean protein content for the milk samples from the grass-fed cattle you collected was greater than the sample mean protein content for the milk samples from grain-fed cattle you collected.
d. The population mean protein content of milk from grass-fed cattle is greater than the population mean protein content of milk from grain-fed cattle.
e. There is sufficient evidence at an α=0.05 level to suggest the population mean protein content of milk from grass-fed cattle is equal to the population mean protein content of milk from grain-fed cattle, based on the collected sample.

Answer 1

Based on the test statistic of 1.98, option c is the only statement that presents an observed fact about the sample data; it claims the sample mean protein content for milk from grass-fed cattle was greater than that for grain-fed cattle.

Step-by-step explanation:

The test statistic of 1.98 can provide insights into which hypothesis is supported by the data, considering Farmer Bill's claim that the milk from grass-fed jersey cattle has a higher protein content than that from grain-fed jersey cattle. To determine which statement is correct, we need to compare the test statistic to critical values from the statistical distribution that apply to the specific hypothesis test being conducted. If the test statistic exceeds the critical value at the chosen alpha level (in this case α=0.05), then there is sufficient evidence to reject the null hypothesis in favor of the alternative.

Without additional information, such as degrees of freedom, the precise critical values, or knowing whether this is a one-tailed or two-tailed test, we can provide a general assessment:

• Option a suggests there is sufficient evidence to conclude there is a difference in favor of the grass-fed cattle, but we need to confirm that the test statistic of 1.98 is significant at the chosen alpha level.
• Option b is about confidence intervals and isn't relevant to hypothesis testing directly.
• Option c could be a simple observational result but does not imply significance on its own.
• Option d is making a claim about the population mean which would need statistical significance to support, and thus cannot be confirmed without further information about significance levels.
• Option e is not supported by the test statistic value since it suggests that the means are equal, contrary to the test statistic implying a difference.

In the context provided, statement option c is the only one that is a factual observation made from the sample data and does not rely on hypothesis testing or significance levels. It simply states what was observed in the sample, without making inferences about the population mean.