Math Problem Statement

The owner of a chain of mini-markets wants to compare the sales performance of two of her stores, Store 1 and Store 2. She records their daily sales on 10 random days and calculates the differences (Store 1 - Store 2). At a 0.05 level of significance, can she conclude that the mean daily sales of Store 2 exceed those of Store 1?

Solution

Let's analyze the provided image step-by-step to solve the problem. Here's a breakdown:

  1. Hypothesis Setup:
    The question involves testing whether the mean daily sales of Store 2 exceed those of Store 1. This is a one-tailed hypothesis test.

    • Null Hypothesis (H0H_0): μd0\mu_d \leq 0 (the population mean of the differences d=Store 1 - Store 2d = \text{Store 1 - Store 2} is less than or equal to zero).
    • Alternative Hypothesis (H1H_1): μd>0\mu_d > 0 (the population mean of the differences is greater than zero).
  2. Data:
    The differences (d=Store 1 - Store 2d = \text{Store 1 - Store 2}) are already calculated in the table: 55,19,157,163,183,160,77,54,144,7955, -19, 157, -163, -183, -160, -77, -54, -144, -79.

  3. Performing a Hypothesis Test:
    The process includes:

    • Calculating the sample mean (dˉ\bar{d}) and sample standard deviation (sds_d) of the differences.
    • Using a tt-test formula since the sample size is small (n=10n = 10): t=dˉ0sd/nt = \frac{\bar{d} - 0}{s_d / \sqrt{n}}
    • Comparing the calculated tt-value to the critical tt-value at α=0.05\alpha = 0.05 for n1=9n-1 = 9 degrees of freedom.

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
t-Test
Descriptive Statistics

Formulas

t = (x̄ - μ) / (s / √n)
x̄ = Σx / n
s = √[Σ(x - x̄)² / (n - 1)]

Theorems

Student's t-Distribution

Suitable Grade Level

Grades 11-12, College Level