91Ó°ÊÓ

Nationally, about \(11 \%\) of the total U.S. wheat crop is destroyed each year by hail (Reference: Agricultural Statistics, U.S. Department of Agriculture). An insurance company is studying wheat hail damage claims in Weld County, Colorado. A random sample of 16 claims in Weld County gave the following data (\% wheat crop lost to hail). $$\begin{array}{cccccccc} 15 & 8 & 9 & 11 & 12 & 20 & 14 & 11 \\ 7 & 10 & 24 & 20 & 13 & 9 & 12 & 5 \end{array}$$ The sample mean is \(\bar{x}=12.5 \% .\) Let \(x\) be a random variable that represents the percentage of wheat crop in Weld County lost to hail. Assume that \(x\) has a normal distribution and \(\sigma=5.0 \% .\) Do these data indicate that the percentage of wheat crop lost to hail in Weld County is different (either way) from the national mean of \(11 \% ?\) Use \(\alpha=0.01\).

Step by step solution

01

Define the Hypotheses

We will use a hypothesis test to determine if the percentage of wheat crop lost to hail in Weld County is different from the national average. Let the null hypothesis (H0) be that the mean percentage loss in Weld County is equal to the national mean: \[ H_0: \mu = 11\% \]The alternative hypothesis (Ha) is that the mean percentage loss in Weld County is different from the national mean:\[ H_a: \mu eq 11\% \]

02

Identify the Test Statistic

Since we have a sample mean and known population standard deviation, we can use the z-test for the hypothesis. The test statistic is calculated using the formula:\[ z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \]where:- \(\bar{x} = 12.5\%\) is the sample mean,- \(\mu = 11\%\) is the population mean,- \(\sigma = 5.0\%\) is the population standard deviation,- \(n = 16\) is the sample size.

03

Calculate the Test Statistic

Substitute the known values into the z-test formula:\[ z = \frac{12.5 - 11}{\frac{5.0}{\sqrt{16}}} = \frac{1.5}{1.25} = 1.2 \]So, the calculated z-statistic is 1.2.

04

Determine the Critical Value and Decision Rule

For a two-tailed test at \(\alpha = 0.01\), we look up the critical z-values. The critical z-values for \(\alpha = 0.01\) are approximately \(-2.576\) and \(2.576\). The decision rule is: - Reject the null hypothesis if \(z\) is less than \(-2.576\) or greater than \(2.576\). - Do not reject the null hypothesis if \(-2.576 \leq z \leq 2.576\).

05

Make a Decision

The calculated z-value of 1.2 does not fall in the critical region, as it is between \(-2.576\) and \(2.576\). Therefore, we do not reject the null hypothesis.

06

Conclusion

At a significance level of \(\alpha = 0.01\), there is not enough evidence to conclude that the percentage of wheat crop lost to hail in Weld County is different from the national mean of 11%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Z-test

The Z-test is a statistical method used to determine if there's a significant difference between the sample mean of a population and a known population mean. It is particularly useful when you have a known population standard deviation. In the context of the wheat crop hail damage, the Z-test helps us decide if the mean percentage of crop loss in Weld County deviates from the national mean of 11%.

To perform a Z-test, we calculate a test statistic, which is the ratio of the difference between the sample mean and the population mean to the standard error of the mean. This ratio tells us how many standard deviations the sample mean is from the population mean. The formula is given by:

• z = \( \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \)
• \( \bar{x} \) is the sample mean
• \( \mu \) is the population mean
• \( \sigma \) is the population standard deviation
• \( n \) is the sample size

The result is then compared against critical values in a Z-table to determine if we should accept or reject the null hypothesis.

Standard Deviation

Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation means that the data points tend to be close to the mean of the dataset, whereas a high standard deviation indicates that the data points are spread out over a wider range.

In hypothesis testing scenarios like the wheat crop example, the standard deviation ( 5.0% ) represents the expected variation in the percentage of wheat crop loss due to hail across the whole country. It is a crucial component in calculating the standard error and thereby the Z-test statistic.

Understanding standard deviation is essential because it helps us evaluate how much observed data deviates from the expected norm, which in turn allows us to test claims about population parameters accurately.

Normal Distribution

The normal distribution is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean. It is often referred to as a "bell curve" because of its shape.

In the context of the exercise, we assume that the percentage of wheat crop loss in Weld County is normally distributed. This assumption is critical as it allows us to apply the Z-test, which relies on normal distribution properties to determine statistical significance.

Key characteristics of a normal distribution include:

• Symmetry: The left and right sides of the distribution are mirror images.
• Mean, median, and mode are all located at the center of the distribution.
• The total area under the curve is equal to 1.

Knowing these properties is important for correctly interpreting the results of statistical tests like the Z-test.

Critical Value

The critical value is the threshold or boundary of the acceptance region for a test statistic in hypothesis testing. When the calculated test statistic exceeds the critical value, we reject the null hypothesis.

In a two-tailed Z-test at a significance level of \( \alpha = 0.01 \), the critical values are approximately \(-2.576\) and \(2.576\). These values tell us the points beyond which the test statistic would indicate a statistically significant deviation from the null hypothesis mean. If the test statistic falls within these boundaries, as with 1.2 in the wheat crop example, the null hypothesis is not rejected.

• Critical values depend on the chosen significance level (\( \alpha \)).
• Lower \( \alpha \) leads to more stringent critical values, implying a smaller probability of rejecting the null hypothesis falsely.
• It is important to understand the role of critical values for making informed decisions based on statistical tests.

Accurately determining and interpreting critical values helps define the reliability and outcome of the statistical inference.