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In hypothesis testing, a two-tailed test is essential to understand when you're assessing whether a parameter, such as a mean, is different from a specified value. In our example, the null hypothesis (

• \(H_0: \mu = 7\)
• Alternative hypothesis: \(H_1: \mu eq 7\)

A two-tailed test is used because we are interested in determining if the true mean is either smaller or larger than 7, without specifying the direction.For a two-tailed test, we consider both extremities of the distribution. When looking up p-values, the z-score tells us how many standard deviations away our result is from the mean. The p-value calculation involves finding the probability of the test statistic, or any more extreme value, occurring under the null hypothesis. To find the p-value, we look at the areas in both tails of the normal distribution. Each tail can help us understand the possibility of extremes in both directions.

A p-value is a probability measure that assesses how well the sample data confirms a hypothesis. In hypothesis testing, it's the probability of observing an effect at least as extreme as the test statistic, assuming the null hypothesis is true.
To calculate the p-value in a two-tailed test, follow these simple steps:

• Find the area under the standard normal distribution curve that's more extreme than the calculated z-score.
• This area is often found using standard z-tables or statistical software.
• Since you're conducting a two-tailed test, double the single-tail area to find the p-value for both extremes.

Let's consider the given exercise:
- For \( z_{0} = 2.05 \), the single-tail p-value is 0.0202. Hence, the two-tailed p-value is \( 2 \times 0.0202 = 0.0404 \).- For \( z_{0} = -1.84 \), the single-tail p-value is approximately 0.0329. Doubling gives \( 2 \times 0.0329 = 0.0658 \).- For \( z_{0} = 0.4 \), this means \( 2 \times 0.3446 = 0.6892 \).When a p-value is low (often below 0.05), this indicates the observed result is significantly different from what the null hypothesis predicts.

The z-score is a statistical measure that describes how many standard deviations an element is from the mean. It's a key part of hypothesis testing and plays a crucial role in deciding whether to reject the null hypothesis.
In the context of our exercise, the z-scores (

• \(z_{0}=2.05\)
• \(z_{0}=-1.84\)
• \(z_{0}=0.4\)

) represent the difference between the sample mean and the hypothesized population mean, scaled by the standard deviation. A higher absolute value of the z-score indicates a more extreme result relative to the null hypothesis.- A z-score of 2.05 means the sample mean is 2.05 standard deviations above the population mean.- A z-score of -1.84 reflects the sample mean being 1.84 standard deviations below the population mean.- A z-score of 0.4 implies the sample mean is close to the population mean, just 0.4 standard deviations away.Knowing the z-score allows you to calculate the p-value, helping determine whether any observed differences are statistically significant.