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When you conduct a survey or an experiment and need to determine how many people have a particular trait or opinion, you use the sample proportion. In this example, your focus is on the number of satisfied customers.To find the sample proportion, use the formula:\[ \hat{p} = \frac{x}{n} \]Where:\

• \( x \) is the number of successes (satisfied customers, which is 850 in this case), and
• \( n \) is the total number of trials or observations (1000 customers surveyed).

Thus, the sample proportion \( \hat{p} \) calculates to \( \frac{850}{1000} = 0.85 \). This value essentially tells us that about 85% of the surveyed customers are satisfied.

The standard error (SE) provides an estimate of how much your sample statistic (like the sample proportion) is expected to vary from the actual population parameter. It's like taking the pulse on how spread out your data points might be.Here, the standard error for a sample proportion is calculated by:\[ SE = \sqrt{\frac{p(1-p)}{n}} \]In our case, the calculations involve:

• \( p \) = 0.9 (the hypothesized true population proportion), and
• \( n \) = 1000 (total surveyed population).

Thus, the standard error becomes:\[ SE = \sqrt{\frac{0.9 \times 0.1}{1000}} = 0.0095 \]This standard error indicates the level of variability one can expect when estimating the sample proportion from the entire population.

A confidence interval offers a range where the true population parameter (like the proportion of satisfied customers) might fall. It gives a context to the sample data, ensuring we can say: "We are 95% confident that the true proportion of satisfied customers is somewhere within this range."To construct a confidence interval for the proportion:\[ \hat{p} \pm z_{\alpha/2} \times SE \]So:

• \( \hat{p} \) = 0.85 (sample proportion),
• \( z_{\alpha/2} \) = 1.96 (z-value from the standard normal distribution for 95% confidence), and
• \( SE \) = 0.0095 (from the previous calculation).

Carry out the calculation:\[ 0.85 \pm 1.96 \times 0.0095 \]This results in the range \[0.8314, 0.8686\]\. Therefore, you can confidently say the actual proportion of satisfied customers lies within this interval with a 95% surety.

The test statistic helps determine if there is a significant difference between the observed sample statistic and the hypothesized population parameter. It translates your sample data into a single value to test against the null hypothesis.The formula for calculating the test statistic \( z \) is:\[ z = \frac{\hat{p} - p}{SE} \]Where:

• \( \hat{p} \) is the sample proportion (0.85),
• \( p \) is the hypothesized population proportion (0.9), and
• \( SE \) is the standard error (0.0095).

Using the above values, we have:\[ z = \frac{0.85 - 0.9}{0.0095} = -5.26 \]A large absolute value of \( z \) (like -5.26) usually suggests a significant difference, leading us to believe that the sample proportion deviates sufficiently from the hypothesized value. Consequently, the probability value (or \( p \)-value) is very low, suggesting strong evidence against the null hypothesis.