When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.
c. Increase in the population standard deviation

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Understand the formula for the confidence interval: The width of a confidence interval is determined by the formula: \( \text{Width} = 2 \cdot Z \cdot \frac{\sigma}{\sqrt{n}} \), where \( Z \) is the critical value, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

Identify the role of the population standard deviation (\( \sigma \)): The population standard deviation is in the numerator of the formula for the margin of error, which directly affects the width of the confidence interval.

Analyze the effect of increasing \( \sigma \): When \( \sigma \) increases, the value of \( \frac{\sigma}{\sqrt{n}} \) also increases, leading to a larger margin of error.

Relate the margin of error to the width: Since the width of the confidence interval is twice the margin of error, an increase in \( \sigma \) will result in a wider confidence interval.

Conclude the relationship: Therefore, an increase in the population standard deviation will increase the width of the confidence interval, assuming all other factors remain constant.

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Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter. It is expressed with a certain level of confidence, typically 95% or 99%. The width of the interval reflects the uncertainty associated with estimating the parameter; a wider interval indicates more uncertainty.

Population Standard Deviation

The population standard deviation is a measure of the dispersion or spread of a set of values in a population. It quantifies how much individual data points deviate from the population mean. A larger standard deviation indicates greater variability in the data, which can lead to wider confidence intervals when estimating population parameters.

Effect of Standard Deviation on Confidence Interval Width

When the population standard deviation increases, the width of the confidence interval also increases, assuming all other factors remain constant. This is because a larger standard deviation indicates more variability in the data, necessitating a broader range to maintain the same level of confidence in capturing the true population parameter.

When all other quantities remain the same, how does the indicated change affect the width of a confidence interval? Explain.c. Increase in the population standard deviation