Lesson Objectives: Student Will Write Null And Alternative T

Lessonlesson Objectivesstudent Will Write Null And Alternative Hypoth

Lesson Objectives: Student will write null and alternative hypotheses. Student will find critical values for testing a mean or proportion against the population mean or proportion using the appropriate test, based on the sample size. Student will find critical values for testing the difference between two means or two proportions using the appropriate test, based on the sample size. What is a hypothesis? In science, you may have learned that the hypothesis is an educated guess.

In statistics, the same definition carries over but has some different applications. A statistical study is similar to the scientific method. From science, you have learned that the scientific method includes the following steps: 1) Ask a question 2) Do background research 3) Construct a hypothesis 4) Test your hypothesis by doing an experiment 5) Analyze the data and draw a conclusion 6) Communicate the results. In statistics, there are two hypotheses that need to be formed once you have defined the problem and completed background research. One is called the "null" hypothesis, and the other is called the "alternative" hypothesis.

Once the study is conducted, we can reject or fail to reject either of the hypotheses based on the results of the study. The Null Hypothesis The null hypothesis is composed of the fact that there is no effect of the treatment on the subjects in the study. For example, if we were trying to investigate the relationship between two variables, our null hypothesis may state that "there is no relationship between the two variables," or if we are trying to see if a new drug has an effect on weight gain, the null hypothesis may state that "the drug has no effect on the weight gain of the subjects." The null hypothesis is the one that we will fail to reject (accept) unless the data provides convincing evidence that it is false.

The Alternative Hypothesis The alternative hypothesis may be referred to as the opposite of the null hypothesis. For example, if the null hypothesis states that there is no relationship between two variables, then the alternative hypothesis should state that "there is a relationship between the two variables that can be measured." If the null hypothesis states that there is no effect on the subject, then the alternative hypothesis should state that "there is an effect on the subject." We will fail to reject (accept) the alternative hypothesis if and only if the data provides convincing evidence that it is true.

Practice Writing Null and Alternative Hypotheses The hypotheses can be written out in words or we may use mathematical symbols to express the hypothesis. Here are a few examples of how to write the null and alternative hypothesis. The most common symbol for the null hypothesis is H₀, and the most common symbol for the alternative hypothesis is H₁. Let's Practice:

• Case I: An agriculturist is doing a study to determine if a fertilizer has any effect on the average height of 100 apple trees. He knows that the average height of unfertilized apple trees is 10 ft. The average height of the 100 apple trees that were treated with fertilizer is 10.8 feet with a standard deviation of 0.5 ft.

1. Do you think that the fertilizer has an effect on the height of the apple trees?

• Null hypothesis (H₀): The fertilizer has no effect on the height of the apple trees. (Sample mean = Population mean)
• Alternative hypothesis (H₁): The fertilizer does have an effect on the height of the apple trees. (Sample mean ≠ Population mean)

• Null hypothesis (H₀): The fertilizer does not make the apple trees taller. (Sample mean = Population mean)
• Alternative hypothesis (H₁): The fertilizer does make the apple trees taller. (Sample mean > Population mean)

It is crucial when writing and testing hypotheses to consider the research question, as this helps shape the correct hypotheses. Choosing the appropriate test involves deciding whether to use a z-test or t-test, which depends on the sample size, and whether to perform a one-tailed or two-tailed test, based on the research hypothesis. Finding critical values involves calculating the test statistic (z or t), then comparing it to a critical value determined by the significance level and the type of test.

For example, testing a mean against a population mean involves stating hypotheses, calculating the test statistic assuming the null hypothesis is true, and then determining whether this falls into the rejection region based on the critical value. Similar procedures are used for testing proportions, differences of means, or differences of proportions, adjusting the formulas accordingly.

Paper For Above instruction

Hypothesis testing is a fundamental aspect of inferential statistics, allowing researchers to make informed conclusions about populations based on sample data. Two key hypotheses—null and alternative—serve as the foundation for this process. Proper formulation of these hypotheses, combined with selecting the appropriate tests, critical values, and understanding the structure of the data, ensures valid and accurate results.

The null hypothesis (H₀) generally posits that there is no effect or no difference—effectively serving as a default or status quo assumption. For example, in investigating whether a new drug impacts weight, the null hypothesis asserts that the drug has no effect. Conversely, the alternative hypothesis (H₁) proposes that there is a measurable effect or difference. This opposition guides the direction of the test and helps determine whether the results are statistically significant.

In practical applications, hypotheses are often expressed in words and symbols. For example, when testing if fertilizer affects the height of apple trees, null and alternative hypotheses can be formalized as follows: H₀: μ = 10 ft versus H₁: μ ≠ 10 ft, where μ represents the population mean height. Deciding whether to conduct a one-tailed or two-tailed test depends on the specific research question; two-tailed tests are common when deviations in either direction are of interest, while one-tailed tests are used when testing for effects in a particular direction.

Choosing the correct test involves assessing the sample size and the data distribution. For large samples (n > 100), a z-test is appropriate, while small samples (n

Critical values serve as thresholds to reject or fail to reject null hypotheses. They are determined based on the significance level (α), often 0.05, and are derived from the standard normal (z) or t distributions. A test statistic exceeding the critical value indicates statistical significance, prompting rejection of the null hypothesis. For instance, in testing if a fertilizer impacts tree height, the calculated z-score would be compared against the critical z-value.

In cases involving proportions, such as defect rates in manufacturing, the normal approximation to the binomial distribution is employed, provided the sample size is sufficiently large. The standard deviation is calculated from the proportion, and the z-score evaluates whether the observed proportion significantly differs from the hypothesized proportion.

When assessing differences between groups—either means or proportions—calculations involve deriving the standard errors and z-scores for the differences. For example, in comparing test scores of students with and without a study guide, the null hypothesis states there is no difference, and the z-score of the difference is computed and compared to the critical value. If the z-score exceeds the critical value, the null hypothesis is rejected, indicating the study guide has a statistically significant effect.

Understanding and correctly applying hypothesis testing principles ensure robust scientific and statistical conclusions. Accurate formulation, selection of tests, and interpretation of critical values underpin the integrity of data analysis and contribute to valid decision-making in research across disciplines.

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