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.

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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.