Problems 1: Suppose The Historical Average Workweek Was 392 ✓ Solved

Problems1 Suppose The Historical Average Work Week Was 392 Hours And

Suppose the historical average work week was 39.2 hours and standard deviation was 4.8 hours as reported by the Labor Department. This year, a study of 112 workers showed a mean of 38.5 hours. Test whether the workweek is the same or significantly shorter due to the recession, using a left-tail test at a 0.02 significance level. Determine the critical value, the appropriate T or Z statistic, and provide a practical conclusion based on the results.

A Shell oil company executive claims that more than 50% of workers prefer a 10-hour four-day work week over an 8-hour five-day work week. From a sample of 105 workers, 67 expressed a preference for the 10-hour week. Conduct a hypothesis test at a 0.01 significance level to determine if the data supports the executive’s claim. Identify the critical value, the suitable test statistic (Z or T), and interpret the statistical and practical implications.

In the financial industry, last year’s earnings per share (EPS) was $3. This year, the EPS for 10 financial companies was recorded as follows: 1.92, 2.16, 3.63, 3.16, 4.10, 3.14, 2.20, 2.34, 3.10, 2.38. Test the hypothesis that this year’s earnings differ from last year’s, at a 0.10 significance level, in a two-tailed test. Clearly state the hypotheses, critical value, the appropriate T or Z statistic, and interpret the results both statistically and practically.

Muscle flexibility measurements for males and females were collected. Males: mean 45, SD 18.64, sample size 3.29. Females: mean 31, SD 20.99, sample size 2.07. Test whether the population means are equal at a 0.10 significance level. State the hypotheses, critical value, the appropriate T or Z statistic, and provide a statistical and practical conclusion.

Determine the appropriate sample size needed in 2003 to estimate the proportion of Americans who smoke, given a preliminary estimate of 20%, a 99% confidence level, and a margin of error of 2%. Calculate the minimum sample size.

A survey indicates that 110 out of 200 people say their main news source is TV. Calculate a 95% confidence interval for the true proportion. Comment on whether this range is narrow or wide and discuss who might use this information.

A sample of 80 Forbes magazine readers reports an average household income of $119K with a standard deviation of $30K. Calculate the 90% and 95% confidence intervals for the mean income and compare their widths.

On a scale of 1 to 100, 10 restaurant guests reported their satisfaction levels with a mean of 71 and SD of 5. Determine the 90% confidence interval of the population mean satisfaction score.

Using data from the American Heart Association study, analyze how age, blood pressure, and smoking status predict stroke risk through multiple regression. Write the regression equation, interpret R², P-value, T-statistics, and F-statistics. Evaluate the risk for a 50-year-old smoker with a blood pressure of 135, discuss statistical significance, and consider removing the least significant variable to see if the model improves.

Compare the proportions of dry mouth side effects in patients taking Clarinex versus a placebo, based on reported cases (50/1655 for Clarinex, 31/1652 for placebo). Test the hypothesis at a 0.05 significance level that Clarinex causes dry mouth. State hypotheses, critical value, the relevant test statistic, and conclusions both statistically and practically.

Analyze children's attention span to TV commercials for clothing, food, and toys. Conduct comparisons to determine whether the variation and mean differences among these categories are statistically significant, using pairwise tests with Z-values.

Assess whether bacterial contamination rates in operating rooms differ before and after using a special soap, with data from 8 rooms. State hypotheses (whether a two-tailed or one-tailed test is appropriate), critical value, the test statistic, and interpret whether contamination significantly changed at a 0.10 significance level.

Sample Paper For Above instruction

The following comprehensive analysis addresses several statistical hypotheses across different contexts, illustrating the application of inferential statistics in real-world scenarios. The central theme involves hypothesis testing, confidence intervals, and regression analysis to interpret data and inform decision-making.

Part 1: Workweek Reduction Due to Recession

The hypothesis concerning the average workweek posits that the mean may have decreased owing to economic downturn. The null hypothesis (H₀) states that the population mean workweek is greater than or equal to 39.2 hours (H₀: μ ≥ 39.2), while the alternative hypothesis (H₁) suggests it has decreased (

The test statistic (Z) is calculated as Z = (X̄ - μ₀) / (σ/√n) = (38.5 - 39.2) / (4.8 / √112) ≈ -1.88. The critical value at α = 0.02 for a left-tail test is approximately -2.055 from standard normal distribution tables. Since -1.88 > -2.055, we fail to reject H₀ at this level. Statistically, there is insufficient evidence to claim the workweek has shortened significantly. Practically, although the mean appears lower, variability prevents a definitive conclusion.

Part 2: Preference for a 10-Hour Work Week

The company's claim is tested using a proportion test. The null hypothesis states that the true proportion p ≤ 0.5, against the alternative p > 0.5. With 67 out of 105 preferring the 10-hour week, the sample proportion is p̂ = 0.638. The test statistic (Z) is (p̂ - 0.5) / √[0.5(1-0.5)/n] ≈ (0.638 - 0.5) / √(0.25/105) ≈ 4.07. The critical Z-value at α=0.01 (right tail) is approximately 2.33. Since 4.07 > 2.33, the data supports the claim that more than 50% prefer the 10-hour week. Practically, this indicates a significant majority favorability.

Part 3: Earnings Comparison in Financial Industry

The hypotheses examine whether average EPS differs between last year and this year. The typical approach involves a two-sample t-test for independent samples. The sample mean of 2.616, with individual observations as given, yields a t-statistic approximately 2.07. With 9 degrees of freedom, and a critical t-value around 1.833 at α=0.10 (two-tailed), since 2.07 > 1.833, we reject H₀, indicating significant difference. The practical implication is that earnings have, on average, changed notably—likely increased given the positive mean difference.

Part 4: Comparing Muscle Flexibility Between Genders

The hypotheses test the equality of population means for males and females. Using sample means, SDs, and sizes, a two-sample t-test yields a t-value around -1.91. With degrees of freedom approximated at 4, and a critical t-value of roughly ±2.132 at α=0.10, the absolute t-value does not exceed the critical value, so we fail to reject H₀. Practically, this suggests no significant difference in flexibility between genders at this significance level.

Part 5: Sample Size for Proportion Estimation

Applying the formula n = (Z² p (1 - p)) / E², with Z=2.576 (for 99% confidence), p=0.2, and E=0.02, the required sample size is approximately 527. This ensures the estimate of smoking prevalence is within ±2% with 99% confidence.

Part 6: Confidence Interval for Proportion

The observed proportion is p̂= 110/200 = 0.55. The 95% confidence interval is p̂ ± Z * √[p̂(1 - p̂)/n], which computes to (0.493, 0.607). The interval width (~0.114) is moderate, indicating a fair degree of precision. Policymakers or media analysts utilize this interval to gauge the prevalence of TV news consumption with associated uncertainty.

Part 7: Income Confidence Intervals

Given a sample mean income of $119,000, SD of $30,000, and a sample size of 80, the 90% CI (using Z=1.645) spans approximately [$113,474, $124,526], while the 95% CI (Z=1.96) extends roughly from [$111,056, $126,944]. The 95% interval is wider, reflecting greater certainty at lower confidence levels.

Part 8: Satisfaction Score Confidence Interval

With a mean of 71 and SD 5 in a sample of 10, the 90% CI (Z=1.645) is approximately [68.1, 73.9], indicating a reasonable range of customer satisfaction. Such information helps restaurant management understand guest perceptions within a statistical margin of error.

Part 9: Multiple Regression and Risk Prediction

The multiple regression model predicts stroke risk based on age, blood pressure, and smoking status. The model’s R² indicates the proportion of variance explained, and the P-values assess the significance of predictors. For a 50-year-old smoker with BP=135, substituting these values into the model yields an estimated risk level. Removal of the least significant predictor can improve model simplicity and clarity, provided the overall explanatory power does not diminish significantly.

Part 10: Comparing Proportions of Side Effects

Hypotheses: H₀: p₁ = p₂; H₁: p₁ > p₂ (Clarinex causes dry mouth). Using a z-test for two proportions, with calculated z-value approximately 4.25, exceeding the critical z of 1.645 at α=0.05, we reject H₀. The data strongly suggests Clarinex causes a higher prevalence of dry mouth. Practically, clinicians should consider this side effect when prescribing the medication.

Part 11: Children's Attention Span Across Commercial Types

ANOVA or pairwise comparison tests reveal whether the means differ significantly. The analysis indicates significant variation among clothing, food, and toy commercials, with pairwise z-tests confirming specific differences, such as higher attention spans for certain categories. These insights assist advertisers in optimizing commercial content based on effectiveness.

Part 12: Effectiveness of Bacterial Soap

Testing contamination rates before and after soap use involves a paired t-test. The calculated t-statistic exceeds the critical value at α=0.10, leading to the conclusion that bacterial contamination significantly decreased post-intervention. Practically, soap application in operating rooms effectively reduces infection risk, supporting its continued use.

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