A customer service company wants to know if their support agents are performing on par with industry standards. According to a report the standard mean resolution time is 20 minutes per ticket. The sample group has a mean at 21 minutes per ticket with a standard deviation of 7 minutes. Can you tell if the company’s support performance is better than the industry standard or not?

To determine if the company's support performance is statistically different from the industry standard, we can conduct a hypothesis test. Since we're comparing a sample mean to a known population mean, a one-sample \( t \)-test is appropriate, assuming the population standard deviation is unknown.

### Steps for Hypothesis Testing:

1. **Null Hypothesis (H₀):**
   - The company's mean resolution time is equal to the industry standard.
   - \( H_0: \mu = 20 \) minutes.

2. **Alternative Hypothesis (H₁):**
   - The company's mean resolution time is different from the industry standard.
   - \( H_1: \mu \neq 20 \) minutes.

3. **Given Data:**
   - Population mean (industry standard), \( \mu = 20 \) minutes.
   - Sample mean, \( \bar{x} = 21 \) minutes.
   - Sample standard deviation, \( s = 7 \) minutes.
   - Sample size, \( n \) (Not given explicitly, so assume or denote as \( n \)).

4. **Test Statistic:**
   The formula for the \( t \)-test statistic is:

   \[
   t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}
   \]

   Plugging in the values:

   \[
   t = \frac{21 - 20}{\frac{7}{\sqrt{n}}} = \frac{1}{\frac{7}{\sqrt{n}}}
   \]

5. **Determine Degrees of Freedom:**
   Degrees of freedom \( df = n - 1 \).

6. **Find the Critical Value or P-value:**
   Based on the calculated \( t \)-value and degrees of freedom, compare this with the critical \( t \)-value from the \( t \)-distribution table (or use the P-value).

7. **Decision:**
   - If the calculated \( t \)-value is greater than the critical \( t \)-value (or the P-value is less than the significance level, typically 0.05), reject the null hypothesis.
   - Otherwise, do not reject the null hypothesis.

To proceed with the calculation, I need the sample size \( n \). Could you provide that information, or would you like to assume a typical value for \( n \)?

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### Follow-up Questions:

1. What is the sample size \( n \) for the company data?
2. Would you like to assume a significance level for the test (e.g., 0.05)?
3. Do you have any additional performance metrics to analyze?
4. Would you prefer using a specific statistical software for the calculation?
5. Is there a need to adjust the performance based on other variables, like ticket complexity?

### Tip:
Always ensure that the sample size is sufficient for hypothesis testing to improve the accuracy and reliability of the results.

Learn how to perform hypothesis testing to determine if a customer service company's mean resolution time is better than the industry standard. Includes step-by-step calculation using a one-sample t-test, with sample mean, standard deviation, and population mean provided.

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