If You Use A 0.10 Level Of Significance In A Two-Tail Hypoth

93if You Use A 010 Level Of Significance In A Two Tail Hypothesis

Use a 0.10 level of significance in a two-tail hypothesis test to determine whether to reject the null hypothesis that the population mean is 500 based on a Z test.

Set up a hypothesis test comparing the mean number of hours students study at your school to the reported 14.6 hours per week from BusinessWeek. Specify the null hypothesis that the mean is equal to 14.6 hours, and the alternative hypothesis that it is different.

Understand the implications of Type I and Type II errors in the context of this hypothesis test: a Type I error involves incorrectly rejecting the null hypothesis when it is true, and a Type II error involves failing to reject the null hypothesis when it is false.

A paint supply store manager tests whether the average amount of paint in 1-gallon cans from a manufacturer is 1 gallon, with known standard deviation 0.02 gallon, based on a sample mean of 0.995 gallon from 50 cans. The significance level is 0.01.

Determine if there's evidence that the true mean differs from 1 gallon, compute the p-value and interpret it, and construct a 99% confidence interval estimate of the population mean. Compare the findings from the hypothesis test and the confidence interval to reach a conclusion.

A savings bank in New York State examines whether the average processing time for approved policies has changed from the historical mean of 45 days. A sample of 27 policies shows total processing times over one month. At a 0.05 significance level, analyze whether the mean processing time has changed.

Discuss the assumptions about the population distribution necessary for conducting the t-test, and evaluate these assumptions by constructing a boxplot or normal probability plot. Provide your opinion on whether these assumptions are valid based on the plots.

Sample Paper For Above instruction

In the realm of hypothesis testing, selecting an appropriate level of significance is crucial for making informed decisions about the null hypothesis. This paper explores several scenarios where a 0.10 significance level is used in a two-tailed test, focusing on the decision rule related to the population mean of 500 based on a Z test, and extends into practical applications involving student study habits, manufacturing quality assessments, and operational efficiency in banking services.

Decision Rule for a Two-Tail Z Test at 0.10 Significance Level

When conducting a two-tail hypothesis test at a 0.10 level of significance, the critical z-values are approximately ±1.645. The null hypothesis (H0) posits that the population mean μ equals 500, while the alternative hypothesis (Ha) states that μ is not equal to 500. The decision rule is: reject H0 if the calculated Z-statistic falls outside the range of -1.645 to 1.645. Conversely, fail to reject H0 if Z falls within this interval. This approach ensures that there is a 10% chance—split evenly between the two tails—of making a Type I error, i.e., incorrectly rejecting the null hypothesis when it is true.

Analyzing Student Study Hours Relative to BusinessWeek Data

To examine whether students at a particular school study more, less, or about the same as students at other top business schools, we set up a hypothesis test: H0: μ = 14.6 hours; Ha: μ ≠ 14.6 hours. Using a sample of student study hours, with the known or estimated standard deviation, we would compute the Z-statistic and compare it to critical values at an appropriate significance level, such as 0.05. The null hypothesis would be rejected if the Z-value exceeds the critical threshold, indicating a statistically significant difference from the benchmark.

Type I error occurs if we wrongly reject H0 when the true mean is indeed 14.6 hours, leading us to falsely conclude that students study differently. Type II error would occur if we fail to reject H0 when in fact the true mean is different, thus missing a real difference.

Evaluating the Mean Amount of Paint in Cans

A paint store manager tests whether the average amount of paint in 1-gallon cans conforms to the promised 1 gallon. With a sample mean of 0.995 gallons, a known standard deviation of 0.02 gallons, and a sample size of 50, we conduct a Z-test at α = 0.01. Calculating the Z-statistic:

Z = (0.995 - 1.0) / (0.02 / √50) ≈ -2.236

Since the critical values at α = 0.01 are approximately ±2.576, the calculated Z doesn’t fall into the rejection region. The p-value corresponding to Z = -2.236 is approximately 0.025, which is greater than 0.01, indicating insufficient evidence to conclude the mean differs from 1 gallon at the 1% significance level.

The 99% confidence interval for the population mean is:

0.995 ± 2.576 * (0.02 / √50) ≈ 0.995 ± 0.0073, resulting in (0.9877, 1.0023) gallons. This interval includes 1 gallon, supporting the conclusion from the hypothesis test that evidence of a difference is weak at this significance level.

Analyzing Processing Time in Savings Banks

The bank's management assesses whether the average processing time for policies has changed from 45 days. Using the sample of 27 policies, total processing days, and a significance level of 0.05, we calculate the sample mean and standard deviation. Assuming the sample mean exceeds or falls below 45, the t-test compares the sample mean to the known average, with degrees of freedom 26.

For the t-test, if the sample mean significantly deviates from 45 days (p-value less than 0.05), we reject the null hypothesis, concluding the processing time has changed. Construction of a boxplot or normal probability plot helps assess the assumption that the population distribution is approximately normal, which is necessary for the validity of the t-test.

Evaluating these plots, if data are roughly symmetric and unimodal, the normality assumption holds reasonably. If distributions are heavily skewed or contain outliers, the assumption may be violated, potentially invalidating the t-test results. Based on the plots, the assumption appears valid if the data sample does not show significant deviations from normality.

This comprehensive analysis underscores the importance of proper hypothesis test setup, understanding of errors, and verification of assumptions in statistical decision-making across various industries and research scenarios.

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