For a normal population with known variance σ^2 : (a) What value of zα/ 2 in Equation 8-5 gives

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For a normal population with known variance σ^2 : (a) What...

For a normal population with known variance $\sigma^{2}$ : (a) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $98 \%$ confidence? (b) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $80 \%$ confidence? (c) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $75 \%$ confidence?

For a normal population with known variance $\sigma^{2}$ :
(a) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $98 \%$ confidence?
(b) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $80 \%$ confidence?
(c) What value of $z_{\alpha / 2}$ in Equation $8-5$ gives $75 \%$ confidence?

Key Concepts

Confidence Interval

A confidence interval is a range of values, derived from a sample, that is believed, with a specified probability (the confidence level), to contain the true population parameter. It provides an estimation framework that quantifies the uncertainty associated with a sample statistic.

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution with a mean of zero and a standard deviation of one. It is widely used in statistical inference, particularly when the population variance is known, as it facilitates the computation of probabilities and the identification of critical values.

Critical Value (z?/2)

A critical value is a point on the distribution curve that is compared to the calculated test statistic to decide whether to reject the null hypothesis. For confidence intervals, z?/2 represents the cutoff point on the standard normal distribution such that the area in the tail beyond this value is equal to ?/2, ensuring that the central area corresponds to the desired level of confidence.

Significance Level

The significance level, denoted as ?, is the probability of rejecting a true null hypothesis. In the context of confidence intervals, it determines the total area of the tails that are not captured by the interval. The choice of confidence level directly impacts the value of ?, and thus the corresponding critical values.

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