Sample Of 47 Observations From A Normal Population

Sample Of 47 Observations Is Selected From A Normal Population Th

A sample of 47 observations is selected from a normal population. The sample mean is 31, and the population standard deviation is 4. Conduct a hypothesis test at the 0.05 significance level to determine whether the population mean exceeds 30. The test involves formulating hypotheses, determining the decision rule, calculating the test statistic, and interpreting the p-value. Additionally, the scenario includes evaluating whether the sample provides sufficient evidence to support claims about the population mean based on sample data and significance levels.

Sample Paper For Above instruction

Introduction

Statistical hypothesis testing is a fundamental process used to make inferences or draw conclusions about a population based on sample data. In this context, we examine a scenario where a sample is taken from a normal population, with the primary goal of testing if the population mean exceeds a specified value. This paper explores the steps involved in hypothesis testing, including formulating hypotheses, selecting the appropriate test, determining decision rules, calculating the test statistic, and interpreting the p-value. Furthermore, real-world applications of hypothesis testing in business, social sciences, and policy decision-making are discussed to illustrate its significance and utility.

Formulating the Hypotheses

The initial step involves establishing null and alternative hypotheses. In this scenario, the null hypothesis (H₀) states that the population mean is less than or equal to 30 (H₀: μ ≤ 30), representing the status quo or no effect. The alternative hypothesis (H₁) posits that the population mean is greater than 30 (H₁: μ > 30), indicating a potential increase or effect. This setup defines a one-tailed test because the alternative hypothesis specifies a directional change—greater than 30.

Determining the Decision Rule

Next, the decision rule is formulated based on the significance level, α = 0.05. Since the population standard deviation is known, we employ a z-test. The critical value (c) corresponds to the z-score where the area to the right under the standard normal curve is 0.05. Using standard z-tables, c ≈ 1.645. The decision rule is: reject H₀ if the calculated z-test statistic exceeds 1.645. Otherwise, we do not have sufficient evidence to reject H₀ at the 5% significance level.

Calculating the Test Statistic

The test statistic for a population mean with known standard deviation is calculated as:

Z = (x̄ - μ₀) / (σ / √n)

where x̄ = 31 (sample mean), μ₀ = 30 (hypothesized mean), σ = 4 (population standard deviation), and n = 47 (sample size). Plugging in the values:

Z = (31 - 30) / (4 / √47) ≈ 1 / (4 / 6.855) ≈ 1 / 0.584 ≈ 1.711

Thus, the calculated z-value is approximately 1.71.

Decision Regarding the Null Hypothesis

Since 1.71 > 1.645, the test statistic exceeds the critical value. Therefore, we reject the null hypothesis at the 0.05 significance level. This indicates that there is statistically significant evidence to support the claim that the population mean exceeds 30.

P-Value Interpretation

The p-value associated with a z of 1.71 can be found using standard normal distribution tables or statistical software. The p-value is approximately 0.0439, which is less than the significance level of 0.05. Since the p-value is less than α, this further reinforces the decision to reject the null hypothesis, providing additional evidence that the true population mean is greater than 30.

Conclusion

Through hypothesis testing, we have found sufficient evidence at the 5% significance level to conclude that the population mean is greater than 30 based on the sample data. This demonstrates the practical utility of hypothesis testing in making informed decisions based on sample statistics, especially when assessing claims or hypotheses about population parameters.

References

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