Sample Of 37 Observations From A Normal Population

A Sample Of 37 Observations Is Selected From A Normal Population The

A sample of 37 observations is selected from a normal population. The sample mean is 29, and the population standard deviation is 5. Conduct the following hypothesis test using the 0.05 significance level:

  • Null hypothesis: H₀ : μ ≤ 26
  • Alternative hypothesis: H₁ : μ > 26

This is a one-tailed test because the alternative hypothesis specifies a direction (greater than). We are testing whether the population mean is greater than 26.

To perform the test, we first calculate the z test statistic using the formula:

z = (x̄ - μ₀) / (σ / √n)

where:

  • x̄ = 29 (sample mean)
  • μ₀ = 26 (hypothesized population mean)
  • σ = 5 (population standard deviation)
  • n = 37 (sample size)

Calculating:

z = (29 - 26) / (5 / √37) ≈ 3 / (5 / 6.08276) ≈ 3 / 0.8216 ≈ 3.649

Next, we determine the critical value for a one-tailed z-test at α = 0.05. From z-tables, the critical value is approximately 1.6449.

Since the calculated z (3.649) > critical value (1.6449), we reject the null hypothesis, indicating sufficient evidence to conclude that the population mean exceeds 26 at the 5% significance level.

Now, we find the p-value associated with z = 3.649. The p-value for z = 3.649 in a one-tailed test is approximately 0.00013 (rounded to four decimal places).

Paper For Above instruction

The hypothesis testing procedure provides a systematic way to determine whether the evidence from sample data supports a specific claim about a population parameter. In this case, the claim under investigation was whether the population mean exceeds 26, based on a sample of 37 observations with a mean of 29 and a known population standard deviation of 5. The test was structured as a one-tailed z-test, appropriate because the population standard deviation was known and the alternative hypothesis specified a direction (greater than).

The calculation of the test statistic revealed a Z-value of approximately 3.649, which significantly exceeds the critical value of 1.6449. This indicates that the observed sample mean is unlikely under the null hypothesis that μ ≤ 26. The null hypothesis was therefore rejected at the 0.05 significance level, implying that there is statistically significant evidence to infer that the true population mean exceeds 26.

The p-value obtained was approximately 0.00013, reinforcing the conclusion that the likelihood of observing such a sample mean if the true population mean were less than or equal to 26 is exceedingly small. Consequently, the data strongly support the alternative hypothesis that the population mean is greater than 26.

This analysis exemplifies how hypothesis testing enables researchers to make informed decisions about population parameters. It also emphasizes the importance of choosing the correct test type (z-test vs. t-test) based on known parameters and the directionality indicated by the research question. When properly applied, hypothesis tests like this help avoid subjective conclusions and ensure that decisions are based on statistically rigorous criteria.

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