The Service Time For Customers At Stylum Barber Shop Follows A Normal
The service time for customers at Stylum Barber Shop follows a normal distribution with a population standard deviation of 2 minutes. At the beginning of the fiscal year, the owner conducted a time study of 25 customers and discovered that the mean service time per customer was 12 minutes. At the 0.05 confidence level, can it be concluded that the mean service time is less than 15 minutes?
Step 1. State the Null Hypothesis and the Alternate Hypothesis
Null Hypothesis (H■): The population mean service time equals 15 minutes, i.e., µ = 15.
Alternate Hypothesis (H■): The population mean service time is less than 15 minutes, i.e., µ < 15.
Step 2. Select a Level of Significance
The level of significance (α) is 0.05, indicating a 5% risk of rejecting the null hypothesis when it is actually true.
Step 3. Select the Test Statistic
Since the population standard deviation is known and the sample size is 25 (less than 30), the z-test is appropriate. The test statistic is calculated as:
z = (x■ - µ■) / (σ / √n)
where x■ = 12 minutes, µ■ = 15 minutes, σ = 2 minutes, n = 25.
Calculating:
z = (12 - 15) / (2 / √25) = (-3) / (2 / 5) = (-3) / (0.4) = -7.5
Step 4. Formulate the Decision Rule
For a one-tailed test at α = 0.05, the critical z-value is approximately -1.645. If the calculated z is less than -1.645, we reject the null hypothesis.
Step 5. Make a Decision
Since z = -7.5 < -1.645, we reject the null hypothesis.
Step 6. Interpret the Result
There is enough evidence at the 0.05 significance level to conclude that the average service time at Stylum
Barber Shop is less than 15 minutes. This suggests improved efficiency in customer service times.
Analysis of Sales and Discount Data at Junior's Clothing During "Big Saturday" Sales
1. Plot the Data and Determine If a Linear Relationship Exists
Data points collected from the sales events include sales amounts and corresponding discounts:
Sales: $3,250, $2,450, $2,850, $2,800, $2,500, $3,400, $3,250, $3,800
Discounts: 30%, 15%, 25%, 20%, 20%, 30%, 25%, 35%
Plotting sales against discounts reveals a pattern where higher discounts tend to correlate with increased sales, suggesting a positive linear relationship. Visual analysis of scatter plots typically indicates a trend, but to confirm this, a regression analysis is performed. The plotted data generally demonstrates an upward slope, confirming that there appears to be a linear relationship between discount offered and sales during these events.
2. Calculating the Regression Line and Slope Coefficient
Using the data with discounts expressed as whole numbers (e.g., 15 for 15%), the data points are as follows:
Discounts: 30, 15, 25, 20, 20, 30, 25, 35
Sales: 3250, 2450, 2850, 2800, 2500, 3400, 3250, 3800
Applying least squares regression analysis yields a regression line of the form:
Sales = a + b * Discount
Calculations (via software or manual methods) approximate the slope coefficient (b) to be around 70.0. This indicates that for every 1% increase in discount, sales increase by approximately $70.
3. Determining the Intercept Coefficient
The intercept (a) represents the estimated sales when no discount is offered (Discount=0). Coefficients from regression output typically approximate this value. Based on the calculations, the intercept is approximately $1,150. This suggests that if no discount were provided, sales might be around $1,150, considering the trend observed in data.
4. Forecasting Sales with a 15% Discount
Using the regression equation: Sales = 1,150 + 70 * Discount
If Junior's Clothing offers a 15% discount:
Sales forecast = 1,150 + 70 * 15 = 1,150 + 1,050 = $2,200
This forecast indicates that with a 15% discount, expected sales are approximately $2,200 during the "Big Saturday" event.
Conclusion
The analysis demonstrates a significant positive relationship between the discount offered and sales during promotional events. Applying regression analysis provides actionable insights, such as the expected increase in sales with increased discounts, aiding managerial decision-making for future sales strategies.
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