A sample of 1000 customers this year resulted in 791 responding favourably
to a certain proposal, if the original last year's customer's favourable response 
to the same proposal was 90% of those responding or P = 0.90, is there a 
change in customer response?

To determine if there has been a statistically significant change in customer response compared to last year's 90% favorable response rate, we can perform a **hypothesis test** for proportions. Specifically, a **z-test for a proportion** can be applied here.

### Step-by-Step Solution:

#### 1. **Define the Hypotheses:**

- **Null Hypothesis (\(H_0\))**: There is no change in customer response. The proportion of favorable responses is still 90%, i.e., \( p = 0.90 \).
- **Alternative Hypothesis (\(H_a\))**: There is a change in customer response, i.e., \( p \neq 0.90 \) (this is a two-tailed test).

#### 2. **Given Data:**
- Sample size: \( n = 1000 \)
- Number of favorable responses: \( x = 791 \)
- Sample proportion: \( \hat{p} = \frac{791}{1000} = 0.791 \)
- Population proportion (last year's proportion): \( p_0 = 0.90 \)

#### 3. **Calculate the Test Statistic (z-value):**

The formula for the z-test statistic for a proportion is:

\[
z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
\]

Substitute the values:

\[
z = \frac{0.791 - 0.90}{\sqrt{\frac{0.90(1 - 0.90)}{1000}}} = \frac{-0.109}{\sqrt{\frac{0.09}{1000}}} = \frac{-0.109}{0.0094868} \approx -11.49
\]

#### 4. **Determine the p-value:**

Since this is a two-tailed test, we need to find the p-value associated with \( z = -11.49 \). A z-value of this magnitude is extremely rare. Looking up the z-value in a standard normal distribution table (or using software), we find that the p-value is almost 0 (far less than 0.05).

#### 5. **Conclusion:**

Since the p-value is very small, we reject the null hypothesis \( H_0 \). This means there is **statistical evidence of a significant change** in customer response compared to last year's 90% favorable rate.

### Final Answer:
Yes, there is a significant change in customer response.

---

Do you want more details about the statistical test or any part of the solution?

Here are 5 related questions to expand on the concept:

1. How would the hypothesis test change if it were one-tailed instead of two-tailed?
2. How is the z-test for proportions different from a t-test?
3. What would happen if the sample size were smaller, say 100 instead of 1000?
4. How would we interpret the result if the p-value was higher, say 0.07?
5. How can confidence intervals be used to further interpret the change in response?

**Tip:** Always check the assumptions of a statistical test before applying it. For a z-test, one key assumption is that the sample size is large enough for the normal approximation to be valid.

A sample of 1000 customers this year resulted in 791 responding favourably to a certain proposal, if the original last year's customer's favourable response to the same proposal was 90% of those responding or P = 0.90, is there a change in customer response?

Explore how to determine if there is a significant change in customer response using a Z-test for proportions. This problem compares a sample response rate of 79.1% from 1000 customers against last year's 90% rate. A step-by-step solution demonstrates hypothesis testing and statistical significance, suitable for Grades 11-12.

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