Answer Using Excel Spreadsheet Must Show All Formulas
Answer Using Excel Spreadsheet Must Show All Formulas915 Suppose Tha
The assignment involves conducting hypothesis tests related to population means and proportions, using different scenarios with specified data and assumptions. The key tasks include reproducing calculations similar to a previous problem (Problem 9.14), applying alternative standard deviations, comparing results, and demonstrating all formulas in an Excel spreadsheet to support decision-making. The scenarios encompass testing CFL bulb lifespan, evaluating delivery times, and analyzing internship data, involving z-tests for means and proportions, calculating p-values, constructing confidence intervals, and interpreting results accordingly.
Paper For Above instruction
This paper aims to methodically analyze various hypothesis testing scenarios using Excel, emphasizing the importance of thorough calculations, formula transparency, and data interpretation. Each scenario mimics real-world decision-making in quality control, service improvement, and survey analysis, providing a comprehensive understanding of statistical inference for different data types and conditions.
Problem 9.14 Recap and Context
The first task addresses evaluating whether the mean lifespan of CFL bulbs is different from 7,500 hours, given a sample mean of 7,250 hours, a population standard deviation of 1,000 hours, and a sample size of 64. The test uses a significance level of 0.05, implying that if the calculated z-statistic exceeds critical z-values, the null hypothesis of the population mean being 7,500 hours will be rejected. The p-value provides an exact probability of observing the data assuming the null hypothesis, while the confidence interval offers a range of plausible population means.
Expanding to a New Standard Deviation
Part a asks to repeat the hypothesis test assuming the population standard deviation is 1,200 hours instead of 1,000. This change impacts the standard error calculation, which influences the z-statistic. The formula for the z-test is:
\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]
where \(\bar{x}\) is the sample mean, \(\mu_0\) the hypothesized mean, \(\sigma\) the population standard deviation, and \(n\) the sample size. By substituting \(\sigma = 1,200\), the standard error increases, potentially affecting the conclusion whether the null hypothesis is rejected.
Comparison of Results
Part b emphasizes comparing outcomes from the original scenario (with \(\sigma = 1,000\)) to the revised scenario with \(\sigma = 1,200\). Changes in the z-statistic, p-value, and confidence intervals are examined. An increased standard deviation generally results in a smaller z-value magnitude and a higher p-value, indicating less evidence to reject the null hypothesis if the data are identical.
Additional Scenarios Involving Hypothesis Tests
Other cases include:
- Testing if delivery times decreased below 25 minutes, using sample data with mean 22.4 minutes, SD 6 minutes, and n=36, applying one-sample z-tests for the mean.
- Evaluating the proportion of unpaid internships, comparing observed proportions to the population proportion (0.48), using z-tests for proportions under different sample outcomes.
Each involves computing test statistics with proper formulas in Excel, visualizing results with p-values and confidence intervals, and interpreting the practical implications.
Excel Implementation and Formulas
In Excel, formulas for these tests include:
- Standard error: `=σ/SQRT(n)`
- Z-statistic for mean: `=(sample_mean - hypothesized_mean) / (σ / SQRT(n))`
- P-value (two-tailed): `=2*NORMSDIST(-ABS(z_value))`
- Confidence interval: `=sample_mean ± (Zcritical * standard_error)`, where `Zcritical` is based on the confidence level (e.g., 1.96 for 95%).
For hypotheses involving proportions, the formulas are:
- Standard error of proportion: `=SQRT((p*(1-p))/n)`
- Z-statistic: `(observed_proportion - p0) / standard_error`
- P-value: similarly calculated using NORMSDIST.
Throughout, all formulas are explicitly calculated in Excel cells to maintain transparency and reproducibility.
Conclusion
Accurate hypothesis testing relies on precise calculation of test statistics, p-values, and confidence intervals. Changes in underlying assumptions, such as the population standard deviation, significantly affect results. Using Excel to perform these calculations ensures clarity and allows for straightforward comparison and interpretation. These practices enhance decision-making processes in quality control, service improvements, and survey analysis, supporting evidence-based conclusions in various real-world applications.
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