Question: market research analysts are making conclusions about the proportion customers willing to add extra services at the low price. They believe that the theoretical proportion may be higher than 25%. A sample of 1,200 respondents was collected. It shows that 333 are willing to add the new services at the low price.

1. Determine the parameters of the normal approximation for the distribution of the sample proportion.
   Mean =
   Standard deviation =
2. At the \(1 \%\) significance level, do analysts have evidence that the proportion of customers willing to add new services exceeded the level of \(25 \%\) ? (yes / no) test statistic = ??
   Standard error = ??
   Critical value = ??
3. State the rejection rule < reject the null hypothesis if... >
4. In the future analysts will design a new study. They intend to estimate the population proportion with the \(98 \%\) confidence. They also want to make sure that the margin of error will not exceed \(4 \%\). How many records should they collect without knowing the sample proportion? Show the critical value for this problem and evaluate the smallest sample size needed for the study.

Critical value =

The smallest sample size =

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