Consider The Following Sample Data On The Age Of The 30 Empl

Consider The Following Sample Data On The Age Of The 30 Emplo

1tco Aconsider The Following Sample Data On The Age Of The 30 Emplo

1tco Aconsider The Following Sample Data On The Age Of The 30 Emplo

1. (TCO A) Consider the following sample data on the age of the 30 employees that were laid off recently from DVC Inc. a. Compute the mean, median, mode, and standard deviation, Q1, Q3, Min, and Max for the above sample data on age of employees being laid off. b. In the context of this situation, interpret the Median, Q1, and Q3. (Points: . (TCO B) Consider the following data on newly hired employees in relation to which part of the country they were born and their highest degree attained. HS BS MS PHD Total East Midwest South West Total If you choose one person at random, then find the probability that the person a. has a PHD. b. is from the East and has a BS as the highest degree attained. c. has only a HS degree, given that person is from the West. (Points: . (TCO B) Squib claims that its new pain reliever is effective in giving relief for headaches within 10 minutes for 95% of users. A random sample of 25 patients is selected. Assuming Squibb is correct, then find the probability that a. exactly 23 patients obtain relief within 10 minutes. b. more than 23 patients obtain relief within 10 minutes. c. at most 22 patients obtain relief within 10 minutes. (Points: . (TCO B) At a local supermarket the monthly customer expenditure follows a normal distribution with a mean of $495 and a standard deviation of $121. a. Find the probability that the monthly customer expenditure is less than $300 for a randomly selected customer. b. Find the probability that the monthly customer expenditure is between $300 and $600 for a randomly selected customer. c. The management of a supermarket wants to adopt a new promotional policy giving a free gift to every customer who spends more than a certain amount per month at this supermarket. Management plans to give free gifts to the top 8% of its customers (in terms of their expenditures). How much must a customer spend in a month to qualify for the free gift? (Points: . (TCO C) A tool manufacturing company wants to estimate the mean number of bolts produced per hour by a specific machine. A simple random sample of 9 hours of performance by this machine is selected and the number of bolts produced each hour is noted. This leads to the following results. Sample Size = 9 Sample Mean = 62.3 bolts/hr Sample Standard Deviation = 6.3 bolts/hr a. Compute the 90% confidence interval for the average number bolts produced per hour. b. Interpret this interval. c. How many hours of performance by this machine should be selected in order to be 90% confident of being within 1 bolt/hr of the population mean number of bolts per hour by this specific machine? (Points: . (TCO C) A clock company is concerned about errors in assembly in their custom made clocks. A simple random sample of 120 clocks yields nine clocks with errors in assembly. a. Compute the 99% confidence interval for the proportion of clocks with errors in assembly. b. Interpret this confidence interval. c. How large a sample size will need to be selected if we wish to have a 99% confidence interval that is accurate to within 1.5%? (Points: . (TCO D) An article in a trade journal reports that nationwide 28% of liquor purchases are made by women. If B & B Liquor’s proportion of sales to women is significantly different from the national norm, the owners are considering redesigning B & B’s advertising. A random sample of 100 customers is selected resulting in 24 women and 76 men. Does the sample data provide evidence to conclude that less than 28% of B&B’s customers are women (using ï¡ = .01)? Use the hypothesis testing procedure outlined below. a. Formulate the null and alternative hypotheses. b. State the level of significance. c. Find the critical value (or values), and clearly show the rejection and non-rejection regions. d. Compute the test statistic. e. Decide whether you can reject Ho and accept Ha or not. f. Explain and interpret your conclusion in part e. What does this mean? g. Determine the observed p-value for the hypothesis test and interpret this value. What does this mean? h. Does this sample data provide evidence (with ï¡ï€ = .01), that less than 28% of B & B’s customers are women? (Points: . (TCO D) Bill Smith is the Worthington Township manager. When citizens request a traffic light, the staff assesses the traffic flow at the requested intersection. Township policy requires the installation of a traffic light when an intersection averages more than 150 vehicles per hour. A random sample of 48 vehicle counts is done. The results are as follows: Sample Size = 48 Sample Mean = 158.3 vehicles/hr. Sample Standard Deviation = 27.6 vehicles/hr. Does the sample data provide evidence to conclude that the installation of the traffic light is warranted (using ï¡ = .10)? Use the hypothesis testing procedure outlined below. a. Formulate the null and alternative hypotheses. b. State the level of significance. c. Find the critical value (or values), and clearly show the rejection and nonrejection regions. d. Compute the test statistic. e. Decide whether you can reject Ho and accept Ha or not. f. Explain and interpret your conclusion in part e. What does this mean? g. Find the observed p-value for the hypothesis test and interpret this value. What does this mean? h. Does this sample data provide evidence (with ï¡ = 0.10), that the installation of the traffic light is warranted? (Points: 24) Question 1. 1. (TCO E) Management at New England Life wants to establish the relationship between the number of sales calls made each week (CALLS, X) and the number of sales made each week (SALES, Y). A random sample of 18 life insurance salespeople were surveyed yielding the data found below. SALES PREDICT Correlations: CALLS, SALES Pearson correlation of CALLS and SALES = 0.956 P-Value = 0.000 Regression Analysis: SALES versus CALLS The regression equation is SALES = - 2.39 + 0.351 CALLS Predictor Coef SE Coef T P Constant -2.392 1.231 -1.94 0.070 CALLS 0...11 0.000 S = 1.50743 R-Sq = 91.5% R-Sq(adj) = 91.0% Analysis of Variance Source DF SS MS F P Regression 1 390.59 390.59 171.89 0.000 Residual Error 16 36.36 2.27 Total 17 426.94 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 15.140 0..315, 15..839, 18..672 1..412, 35..107, 37.237)XX XX denotes a point that is an extreme outlier in the predictors. Values of Predictors for New Observations New Obs CALLS a. Analyze the above output to determine the regression equation. b. Find and interpret β ˆ1in the context of this problem. c. Find and interpret the coefficient of determination (r-squared). d. Find and interpret coefficient of correlation. e. Does the data provide significant evidence (ï¡ï€ = .05) that the number of calls can be used to predict the sales? Test the utility of this model using a two-tailed test. Find the observed p-value and interpret. f. Find the 95% confidence interval for mean sales for all weeks having 50 calls. Interpret this interval. g. Find the 95% prediction interval for the sales for 1 week having 50 calls. Interpret this interval. h. What can we say about the sales when we had 100 calls in a week? (Points : Question 1. 1. (TCO E) Sam Smith, owner and general manager of Campus Stationery Store, is concerned about the sales behavior of a scanner at the store. He understands that there may be many factors, which may help explain sales, but he believes that advertising and price are major determinants of sales. Sam collects the data given below with Y=SALES (# of sales), X1=ADS (# of ads), X2= PRICE ($) SALES ADS PRICE Predict ADS Predict PRICE Regression Analysis: SALES versus ADS, PRICE The regression equation is SALES = 157 + 4.33 ADS - 1.14 PRICE Predictor Coef SE Coef T P Constant 157.50 33.78 4.66 0.002 ADS 4.327 1.078 4.01 0.005 PRICE -1.1428 0..27 0.004 S = 10.1422 R-Sq = 82.9% R-Sq(adj) = 78.1% Analysis of Variance Source DF SS MS F P Regression 2 3502.0 1751.0 17.02 0.002 Residual Error 7 720.1 102.9 Total 9 4222.1 Predicted Values for New Observations New Obs Fit SE Fit 95% CI 95% PI 1 52.20 4.67 (41.16, 63..80, 78.61) Values of Predictors for New Observations New Obs ADS PRICE 1 10.0 130 Correlations: SALES, ADS, PRICE SALES ADS ADS 0.621 0.055 PRICE -0.661 0.008 0.037 0.982 Cell Contents: Pearson correlation P-Value a. Analyze the above output to determine the multiple regression equation. b. Find and interpret the multiple index of determination (R-Sq). c. Perform the t-tests on β ˆ1and on β ˆ2(use two tailed test with (ï¡ï€ = .05). Interpret your results. d. Predict the number of sales given that there were 10 ads and the price was $130. Use both a point estimate and the appropriate interval estimate. (Points : 31)

Paper For Above instruction

The provided data encompasses multiple statistical analyses on different aspects of business and operational metrics, including descriptive statistics, probability, confidence intervals, hypothesis testing, and regression analysis. This paper synthesizes and applies these statistical methods to interpret the data appropriately, drawing meaningful insights relevant to each scenario described.

Part 1: Descriptive Statistics on Employee Age Data

The initial dataset involves the ages of 30 employees who were recently laid off from DVC Inc. This analysis aims to compute key descriptive statistics: mean, median, mode, standard deviation, quartiles (Q1 and Q3), minimum, and maximum. These metrics provide an overview of the central tendency and variability within the dataset. The mean age, calculated as the sum of all ages divided by 30, offers a measure of average employee age. The median, the middle value when data are ordered, indicates the typical employee age. The mode identifies the most common age in the dataset, revealing potential clustering (e.g., age groupings). The standard deviation quantifies dispersion around the mean, reflecting variability in employee ages. Quartiles divide the data into four parts, with Q1 and Q3 representing the 25th and 75th percentiles, respectively, providing insight into the spread and skewness of ages. Minimum and maximum ages further bound the data, indicating the youngest and oldest employees.

Interpreting the median, Q1, and Q3 in context, the median age indicates the central point, suggesting that half of the employees were younger and half older. Q1 reflects the age below which 25% of employees fell, and Q3 indicates the age below which 75% fell. If the median is closer to the lower quartile, the data may be skewed towards younger ages, whereas proximity to the upper quartile suggests a skew towards older ages.

Part 2: Probability in Employee Hiring Data

The dataset describing newly hired employees includes their birthplace regions and highest degree attained—categorized as HS, BS, MS, and PHD. Calculating probabilities involves determining the likelihood of randomly selecting a person with specific characteristics. For example, the probability of choosing an employee with a PhD is computed by dividing the number of PhD holders by the total hires. Similarly, the probability of selecting an employee from the East with a BS degree considers the joint occurrence over the total hires. The probability that a randomly chosen person from the West has only a high school degree involves conditional probability, dividing the number of HS graduates from the West by the total from the West.

Part 3: Binomial Probability in Pain Reliever Effectiveness

Squibb's claim about the effectiveness of its pain reliever with a success rate of 95% is modeled using the binomial distribution, appropriate for fixed number of trials and binary outcomes (relief within 10 minutes or not). For a sample size of 25, the probability of exactly 23 patients experiencing relief is computed using the binomial probability formula. The calculation for more than 23 patients involves summing probabilities from 24 to 25, while at most 22 corresponds to the cumulative probability up to 22, computed via binomial cumulative distribution functions.

Part 4: Normal Distribution in Customer Expenditure

The supermarket's monthly expenditure follows a normal distribution with a known mean ($495) and standard deviation ($121). To find the probability that a randomly selected customer spends less than $300, the z-score is calculated: (300 - 495)/121 ≈ -1.65, and the corresponding probability is obtained from the standard normal table. Similarly, for expenditures between $300 and $600, the z-scores are computed, and the probabilities are determined by the difference of cumulative probabilities. The promotional policy targeting the top 8% of spenders involves finding the expenditure level corresponding to the 92nd percentile of the normal distribution. This is achieved by identifying the z-score associated with 0.92 in the standard normal table—approximately 1.405—and converting back to dollar amount.

Part 5: Estimating Mean Bolts Production per Hour

A sample of 9 hours yields a sample mean of 62.3 bolts and a standard deviation of 6.3 bolts. The 90% confidence interval for the true mean number of bolts produced per hour is computed using the t-distribution: the t-value for 8 degrees of freedom at 90% confidence is approximately 1.860. The margin of error is t*(s/√n). The interval provides a range within which we expect the true mean to fall with 90% confidence, aiding operational decision-making.

Part 6: Confidence Interval for Proportion of Errors in Clocks

From 120 clocks, 9 are with errors, resulting in an observed proportion of 0.075. The 99% confidence interval for this proportion utilizes the z-distribution for large samples to estimate the true error rate. The formula involves the sample proportion ± z(√(p̂(1-p̂)/n)). To achieve a margin of error of ±1.5%, the required sample size is calculated based on the desired precision and confidence level, employing the formula n = (z² p̂ (1 - p̂)) / E².

Part 7: Hypothesis Testing for Liquor Sales and Traffic Data

The analysis tests whether B & B Liquor’s proportion of women customers differs from the national average of 28%, using a z-test for proportions. The null hypothesis states the proportion equals 0.28, while the alternative hypotheses propose a less-than condition. The test statistic is calculated based on the observed proportion (0.24) and the sample size, compared to critical z-values for ï¡ = 0.01. A similar approach assesses whether the average traffic at an intersection warrants a traffic light, with a sample mean exceeding 150 vehicles per hour considered significant at ï¡ = 0.10. In both cases, p-values provide additional evidence regarding hypotheses rejection.

Part 8: Regression Analysis: Predicting Sales from Calls, Ads, and Price

The linear regression between sales and predictors (calls, ads, price) reveals a strong positive correlation (r = 0.956), with the regression equation indicating that each additional call increases sales by approximately 0.351 units. The coefficient of determination (R² = 91.5%) indicates that over 91% of the variation in sales is explained by the number of calls. Hypothesis tests on regression coefficients determine the significance of predictors: with p-values less than 0.05, both calls and ads significantly predict sales, whereas the negative coefficient for price suggests higher prices reduce sales. Confidence intervals for mean and individual predictions for sales at specific levels of calls and ads are derived, enabling operational forecasting and planning.

Part 9: Multiple Regression: Advertising and Price Impact on Sales

The regression model incorporating ads and price predicts sales with a high degree of confidence (R² = 82.9%). T-tests verify that both predictors significantly influence sales. The prediction of sales with 10 ads and loyal pricing approaches provides a point estimate of approximately 52 sales, with intervals quantifying uncertainty. These insights assist in marketing strategy and inventory management, demonstrating the practical utility of multiple regression analysis in business decision-making.

Overall, the analysis emphasizes the importance of descriptive statistics, probabilistic modeling, confidence intervals, hypothesis testing, and regression techniques in translating business data into actionable insights. These methods contribute to optimizing operational processes, marketing strategies, and resource allocation, ultimately supporting data-driven decision-making in business environments.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
• Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
• Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W. H. Freeman.
• Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics