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5.2.1 Confidence Interval for Population Mean

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    5.1.3 Effect of Alpha and Sample Size on Confidence Interval
  - 5.2 Confidence Interval for Population Mean
    5.2 Confidence Interval for Population Mean
    
    5.2.1 Confidence Interval for Population Mean 5.2.1 Confidence Interval for Population Mean
    Table of contents
    
    Confidence Interval When σ (Sigma) is Known
    
    Confidence Interval When σ (Sigma) is Unknown
    
    Determining the Sample Size
    
    Practical Examples
    
    Conclusion
    
    5.2.2 Confidence Interval for Population Mean - 𝜎 Unknown
    
    5.2.3 Examples of Confidence Interval - 𝜎 Unknown
    
    5.2.4 Determining the Sample Size
  - 5.3 Confidence Interval for Proportion
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    5.3.1 Confidence Interval for Proportion
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    5.7.2 Summary - Hypothesis Testing for Population Mean
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  Module 08 Strategizing Promotions
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    8.1.1 Introduction to Promotion
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    8.2.1 Macro Communication
    
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    8.2.5 Developing Effective Ad Campaigns
    
    8.2.6 Ideal Ad Campaigns : Objectives of Ad Communications
    
    8.2.7 Designing Marketing Communications
  - 8.3 Budgeting
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    8.3.1 Budgeting
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    8.4 Personal Selling
    
    8.4.1 Personal Selling
    
    8.4.2 Sales Process in B2B Market
    
    8.4.3 Sales Process in B2C Market
  - 8.5 Sales Force Management
    8.5 Sales Force Management
    
    8.5.1 Designing Effective Sales Force
    
    8.5.2 Managing the Sales Force

5.2.1 Confidence Interval for Population Mean¶

Confidence Interval When σ (Sigma) is Known¶

Overview: Describes constructing confidence intervals when the population standard deviation (σ) is known.
Key Formula: The confidence interval for the population mean (μ) is calculated as: [ \bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right) ] where \( z_{\alpha/2} \) is derived from the standard normal distribution for a given confidence level.

Confidence Interval When σ (Sigma) is Unknown¶

Adjustment: When σ is unknown, the sample standard deviation (s) replaces σ, and the t-distribution is used instead of the normal distribution: [ \bar{x} \pm t_{\alpha/2, n-1} \left(\frac{s}{\sqrt{n}}\right) ] where \( t_{\alpha/2, n-1} \) reflects critical values from the t-distribution with \( n-1 \) degrees of freedom.
Rationale: This adjustment accounts for the additional uncertainty introduced by estimating σ from the sample.

Determining the Sample Size¶

Objective: To calculate the necessary sample size (n) to achieve a desired margin of error (E) for confidence intervals.
Methodology: [ n = \left(\frac{z_{\alpha/2} \sigma}{E}\right)^2 ] This formula assumes σ is known. If σ is unknown, preliminary estimates from earlier studies or a pilot study may be used.

Practical Examples¶

Amit Mishra's Case: A shop owner calculates the 80% and 95% confidence intervals for average customer sales using known σ.
Puneet Prasad's Scenario: A spa owner estimates the average cost of massage sessions using the t-distribution due to an unknown σ.

Conclusion¶

Summary: This module highlights the practical application of constructing confidence intervals under various conditions and emphasizes the importance of selecting an appropriate sample size based on the desired accuracy of the estimates.
Future Directions: Continues with applying these statistical techniques to more complex business decisions, ensuring robust and reliable outcomes from statistical analyses.

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