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Let's address the questions in the image you've provided. I'll break down the tasks as follows:

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### Question 12:

A travel agency call centre is interested in estimating the average number of calls received per day based on a sample of 21 days. The mean and standard deviation of calls received were provided as:

- Sample size (\( n \)) = 21
- Sample mean (\( \bar{x} \)) = 166.2
- Sample standard deviation (\( s \)) = 22.8

The calls are assumed to be normally distributed.

#### a. **95% Confidence Interval for the Mean Number of Daily Calls**
We will use the formula for the confidence interval for the mean when the population standard deviation is unknown (t-distribution):
\[
CI = \bar{x} \pm t_{\alpha/2} \left( \frac{s}{\sqrt{n}} \right)
\]
Where:
- \( t_{\alpha/2} \) is the t-critical value from the t-distribution for \( n-1 \) degrees of freedom and the confidence level.
- \( s \) is the sample standard deviation.
- \( n \) is the sample size.

#### b. **99% Confidence Interval for the Mean Number of Daily Calls**
For the 99% confidence interval, we will use the same formula but with a different \( t_{\alpha/2} \).

#### c. **Estimate Total Number of Calls Over 30 Days with 95% Confidence**
We first estimate the mean total calls for 30 days as \( 30 \times \bar{x} \), and the 95% confidence interval for this total can be found by multiplying both the lower and upper bounds of the daily call confidence interval by 30.

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### Question 13:

The average dividend yield of a sample of 28 BSE-listed companies is analyzed.

- Sample size (\( n \)) = 28
- Sample mean (\( \bar{x} \)) = 12.5%
- Sample standard deviation (\( s \)) = 3.4%

#### **90% Confidence Interval for the Actual Mean Dividend Yield**
Here, the formula remains the same, except we will use a 90% confidence level.

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I'll start solving them based on the formulas mentioned above. Should I proceed with this approach?

A travel agency call centre wants to know the average number of calls received per day based on a sample. A random sample of 21 days is selected, with a sample mean number of 166.2 calls and a standard deviation of 22.8. The calls received daily are assumed to be normally distributed. The problem involves calculating a 95% and 99% confidence interval for the mean, and estimating the total number of calls over 30 days.

Calculate 95% and 99% confidence intervals for the average number of daily calls received by a call center based on a sample. Learn key concepts of the t-distribution, confidence intervals, and how to estimate the total number of calls over a 30-day period.

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