Identifying the Nature of a Hypothesis Test In Exercises 37–42, state and in words and in symbols. Then determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. Explain your reasoning. Sketch a normal sampling distribution and shade the area for the P-value.

Survey A polling organization reports that the number of responses to a survey mailed to 100,000 U.S. residents is not 100,000.

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Step 1: Define the null hypothesis (H₀) and the alternative hypothesis (H₁) in both words and symbols. The null hypothesis (H₀) typically represents the status quo or no change. In this case, H₀: 'The number of responses to the survey is equal to 100,000' (symbolically, H₀: μ = 100,000). The alternative hypothesis (H₁) represents the claim being tested. Here, H₁: 'The number of responses to the survey is not equal to 100,000' (symbolically, H₁: μ ≠ 100,000).

Step 2: Determine the type of hypothesis test. Since the alternative hypothesis (H₁) uses 'not equal to' (≠), this indicates a two-tailed test. A two-tailed test is used when we are interested in deviations in both directions (greater than or less than the hypothesized value).

Step 3: Sketch the normal sampling distribution. Draw a bell-shaped curve to represent the normal distribution. Mark the mean (μ = 100,000) at the center of the curve. Since this is a two-tailed test, shade the areas in both tails of the curve that correspond to the P-value. These shaded areas represent the probability of observing a test statistic as extreme as, or more extreme than, the one calculated from the sample data, under the assumption that H₀ is true.

Step 4: Explain the reasoning for the test type. The reasoning for a two-tailed test is based on the alternative hypothesis (H₁: μ ≠ 100,000). This hypothesis does not specify a direction (greater than or less than), so we are testing for any significant difference from the hypothesized value in either direction.

Step 5: To calculate the P-value, you would first compute the test statistic (e.g., z-score or t-score) using the sample data. Then, use statistical tables or software to find the probability associated with the test statistic in both tails of the distribution. The total P-value is the sum of the probabilities in both tails.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating two competing hypotheses: the null hypothesis (H0), which represents a statement of no effect or no difference, and the alternative hypothesis (H1), which indicates the presence of an effect or difference. The goal is to determine whether there is enough evidence in the sample data to reject the null hypothesis in favor of the alternative.

Types of Hypothesis Tests

Hypothesis tests can be classified as left-tailed, right-tailed, or two-tailed based on the direction of the alternative hypothesis. A left-tailed test is used when the alternative hypothesis states that a parameter is less than a certain value, while a right-tailed test is used when it states that the parameter is greater. A two-tailed test is appropriate when the alternative hypothesis indicates that the parameter is simply different from a certain value, without specifying a direction.

P-value and Normal Distribution

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis. It represents the probability of obtaining a test statistic as extreme as, or more extreme than, the observed value under the assumption that the null hypothesis is true. In the context of a normal distribution, the P-value corresponds to the area under the curve in the tail(s) of the distribution, which is shaded to visually represent the likelihood of observing the sample data if the null hypothesis holds.

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