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Sagot :
To determine whether Kavita should accept or reject the null hypothesis \( H_0 \) that the average receipt for the branch is [tex]$72.00$[/tex], we need to perform a hypothesis test using the following steps:
1. State the Hypotheses:
- Null Hypothesis (\( H_0 \)): \( \mu = 72 \)
- Alternative Hypothesis (\( H_a \)): \( \mu < 72 \)
2. Set the Significance Level:
- Given the critical z-value for a 5% significance level (lower-tail test) is \( -1.65 \).
3. Calculate the Test Statistic:
- Given:
- Chain average (\( \mu \)): 72.00
- Chain standard deviation (\( \sigma \)): 11.00
- Branch average (\( \bar{x} \)): 67.00
- Sample size (\( n \)): 40
- Calculate the standard error (\( SE \)):
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{11.00}{\sqrt{40}} \approx 1.74 \][/tex]
- Calculate the z-statistic:
[tex]\[ z = \frac{\bar{x} - \mu}{SE} = \frac{67.00 - 72.00}{1.74} \approx -2.875 \][/tex]
4. Compare the Test Statistic to the Critical Value:
- The calculated z-statistic is \( -2.875 \).
- The critical value for a 5% significance level is \( -1.65 \).
5. Decision:
- Since \( -2.875 \) is less than \( -1.65 \), we reject the null hypothesis \( H_0 \).
Therefore, Kavita should reject \( H_0: \mu = 72 \) and accept \( H_a: \mu < 72 \). This means that the average customer receipt for the branch is significantly below the chain's average.
The correct choice is:
- She should reject [tex]\( H_0: \mu = 72 \)[/tex] and accept [tex]\( H_a: \mu < 72 \)[/tex].
1. State the Hypotheses:
- Null Hypothesis (\( H_0 \)): \( \mu = 72 \)
- Alternative Hypothesis (\( H_a \)): \( \mu < 72 \)
2. Set the Significance Level:
- Given the critical z-value for a 5% significance level (lower-tail test) is \( -1.65 \).
3. Calculate the Test Statistic:
- Given:
- Chain average (\( \mu \)): 72.00
- Chain standard deviation (\( \sigma \)): 11.00
- Branch average (\( \bar{x} \)): 67.00
- Sample size (\( n \)): 40
- Calculate the standard error (\( SE \)):
[tex]\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{11.00}{\sqrt{40}} \approx 1.74 \][/tex]
- Calculate the z-statistic:
[tex]\[ z = \frac{\bar{x} - \mu}{SE} = \frac{67.00 - 72.00}{1.74} \approx -2.875 \][/tex]
4. Compare the Test Statistic to the Critical Value:
- The calculated z-statistic is \( -2.875 \).
- The critical value for a 5% significance level is \( -1.65 \).
5. Decision:
- Since \( -2.875 \) is less than \( -1.65 \), we reject the null hypothesis \( H_0 \).
Therefore, Kavita should reject \( H_0: \mu = 72 \) and accept \( H_a: \mu < 72 \). This means that the average customer receipt for the branch is significantly below the chain's average.
The correct choice is:
- She should reject [tex]\( H_0: \mu = 72 \)[/tex] and accept [tex]\( H_a: \mu < 72 \)[/tex].
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