Question

For each of the following confidence levels, determine the corresponding \(Z\) score. $$ \begin{array}{cccc} \hline \text { Confidence Level } & \text { Alpha } & \text { Area Beyond } Z & Z \text { score } \\ \hline 95 \% & 0.05 & 0.0250 & \pm 1.96 \\ 94 \% & & & \\ 92 \% & & & \\ 97 \% & & & \\ 98 \% & & & \\ \hline \end{array} $$

Short answer

94%: ±1.88, 92%: ±1.75, 97%: ±2.17, 98%: ±2.33.

Step-by-step solution

Understanding the Confidence Level and Alpha

The confidence level represents the percentage of all possible samples that can be expected to include the true population parameter. Alpha (α) is the level of significance, which is 1 minus the confidence level. For example, for a 95% confidence level, α = 1 - 0.95 = 0.05.

Splitting Alpha for Two-Tailed Test

For a two-tailed test, the alpha level is split between the two tails of the normal distribution. This means each tail will have an area of α/2. For the 95% confidence level, each tail will have 0.025 (0.05/2) beyond the Z score.

Finding the Z score Using Z Tables

The Z score can be found using standard Z tables or a calculator with inverse normal distribution functions. The value we look up corresponds to the area in one tail.

Calculation for Different Confidence Levels

- **94% Confidence Level**: α = 0.06, α/2 = 0.03. The Z score is approximately ±1.88 (look up 0.03 area in Z table).- **92% Confidence Level**: α = 0.08, α/2 = 0.04. The Z score is approximately ±1.75 (look up 0.04 area in Z table).- **97% Confidence Level**: α = 0.03, α/2 = 0.015. The Z score is approximately ±2.17 (look up 0.015 area in Z table).- **98% Confidence Level**: α = 0.02, α/2 = 0.01. The Z score is approximately ±2.33 (look up 0.01 area in Z table).

Complete the Table

Fill in the values in the table based on the calculations:| Confidence Level | Alpha | Area Beyond Z | Z score | |----------------|-------|---------------|-----------| | 95% | 0.05 | 0.0250 | ±1.96 | | 94% | 0.06 | 0.0300 | ±1.88 | | 92% | 0.08 | 0.0400 | ±1.75 | | 97% | 0.03 | 0.0150 | ±2.17 | | 98% | 0.02 | 0.0100 | ±2.33 |

Key concepts

Z score

A Z score, also called a standard score, tells you how many standard deviations an element is from the mean. It is used in statistics to determine the relative position of a value within a normal distribution. The Z score formula is:
Z = (X - μ) / σ
where:

• X is the value
• μ is the mean of the distribution
• σ is the standard deviation of the distribution

For a 95% confidence level, the Z score is approximately ±1.96, which means our result lies within 1.96 standard deviations of the mean.

alpha level

The alpha level (α) is the threshold for significance in hypothesis testing; it represents the probability of rejecting a true null hypothesis. Also known as the significance level, alpha is typically set at 0.05, but it can vary. To find the alpha level:
Alpha = 1 - Confidence Level
For example, for a 95% confidence level, α = 1 - 0.95 = 0.05. An alpha level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

normal distribution

A normal distribution, also known as a Gaussian distribution, is a probability distribution that is symmetric around the mean. The shape of this distribution is often referred to as a 'bell curve' because of its distinctive appearance. Key properties include:

• The mean, median, and mode are all equal
• It is perfectly symmetrical
• The total area under the curve is 1

Most data points lie close to the mean, creating a peak at the centre and tails that decrease rapidly. Z scores are calculated assuming a normal distribution. This makes it easier to interpret the data.

two-tailed test

In a two-tailed test, the area of the significance level is split between both tails of the normal distribution. This type of test checks for the possibility of an effect in two directions, positive and negative. Therefore, for a 95% confidence level:
Alpha (α) = 0.05
Alpha/2 = 0.025 (0.05/2)
This means that both tails of the distribution will have 2.5% of the total area beyond the Z score. Thus, if your Z score lies in either tail beyond α/2, you reject the null hypothesis.

significance level

The significance level is another term for alpha (α), representing the probability of making a Type I error – that is, rejecting a true null hypothesis. In the context of a confidence level:

• A lower alpha level means stronger evidence is required to reject the null hypothesis
• Common levels are 0.05, 0.01, and 0.10

For instance, with a 98% confidence level, the significance level is α = 0.02. The choice of significance level depends on the field of study, with more critical fields like medicine often using stricter levels.