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The z-test is a type of hypothesis test that helps us determine if there is a significant difference between sample and population means. It's commonly used when the population standard deviation is known and the sample size is sufficiently large, typically over 30. However, it can also be used with smaller samples if the data is normally distributed.

In this scenario, the z-test is applied because the population standard deviation is given as 7 acres, and even though our sample size of 22 farms is below 30, the problem indicates that the variable is normally distributed. When conducting a z-test, we calculate a z-score, which tells us how many standard deviations our sample mean is from the population mean.

• Z-test requirements include knowing the population standard deviation.
• It is applicable for samples where the distribution is approximately normal.
• The test checks if the sample mean significantly differs from the population mean.

The formula for calculating the z-score in a z-test is: \[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean, \( \sigma \) is the population standard deviation, and \( n \) is the sample size. This formula helps in determining the position of the sample mean within the population distribution.

The p-value is a measure that helps us determine the strength of the evidence against the null hypothesis. It indicates the probability of observing the sample data, or something more extreme, assuming that the null hypothesis is true.

In essence, the p-value quantifies the evidence and helps decide whether to reject the null hypothesis. A lower p-value suggests stronger evidence against the null hypothesis. In hypothesis testing, we compare the p-value to a significance level \( \alpha \) (often 0.05 or 0.10) to make a decision.

• If the p-value is less than \( \alpha \), we reject the null hypothesis.
• If the p-value is greater than \( \alpha \), we fail to reject the null hypothesis.

For this exercise, the calculated p-value is approximately 0.2052. This value is more than the chosen \( \alpha = 0.10 \), and thus, we do not have enough evidence to reject the null hypothesis. This means the data does not support a significant change in farm acreage from the previous average of 65 acres.

The null hypothesis, denoted as \( H_0 \), is a statement that there is no effect or no difference. It serves as the starting assumption in a hypothesis test. In statistical testing, the null hypothesis typically states that something is equal to a standard or expected value.

For the given problem, the null hypothesis is \( H_0: \mu = 65 \), which means that the average acreage of farms has not changed and remains at 65 acres. Hypothesis tests aim to gather evidence against \( H_0 \) and show that an alternative explanation, the alternative hypothesis, might be true.

• \( H_0 \) provides a baseline or reference point for testing.
• It is assumed true unless evidence strongly indicates otherwise.
• Failing to reject \( H_0 \) does not prove it true, just insufficient evidence to discount it.

In hypothesis testing, do not equate a failure to reject \( H_0 \) with proving that \( H_0 \) is true. It merely indicates that there isn't enough statistical evidence to support a change or effect.

The alternative hypothesis, represented by \( H_1 \), is a statement that contradicts the null hypothesis. It proposes that there is indeed a significant effect or difference present. In testing, if sufficient evidence suggests that the null hypothesis is unlikely, the alternative hypothesis may be accepted.

In this specific example, the alternative hypothesis is \( H_1: \mu eq 65 \). It postulates that the average acreage of farms has changed from 65 acres, which could mean either an increase or decrease. This makes it a two-tailed test, where potential outcomes on both sides of the mean are considered.

• \( H_1 \) provides an alternative to \( H_0 \) in hypothesis testing.
• It suggests that a real change or difference exists.
• Accepting \( H_1 \) requires solid evidence against \( H_0 \).

Determining whether \( H_1 \) is accepted depends on comparing the calculated p-value with the significance level \( \alpha \). If the null hypothesis is rejected, it suggests that the data supports some difference or change, aligned with \( H_1 \).