Z-table

Definition

The z-table, also known as the standard normal distribution table, is a statistical tool used to find the probability or area under the standard normal curve for a given z-score. It is a crucial component in hypothesis testing and comparing two independent population proportions, as it helps determine the likelihood of observed sample statistics occurring under the null hypothesis.

5 Must Know Facts For Your Next Test

1. The z-table provides the cumulative probability or area under the standard normal curve to the left of a given z-score.
2. The z-table is used to calculate p-values, which represent the probability of obtaining a test statistic as extreme or more extreme than the one observed, given that the null hypothesis is true.
3. In hypothesis testing, the z-table is used to determine critical values for the test statistic, which are then compared to the observed test statistic to make a decision about the null hypothesis.
4. When comparing two independent population proportions, the z-table is used to calculate the test statistic and the corresponding p-value to determine if the difference between the two proportions is statistically significant.
5. The z-table is a standardized reference table, meaning that it can be used for any normal distribution by first standardizing the data to a z-score.

Review Questions

• Explain how the z-table is used in the context of a full hypothesis test example.
  • In a full hypothesis test, the z-table is used to determine the critical value(s) for the test statistic, which is typically a z-score. The observed test statistic is then compared to the critical value(s) to decide whether to reject or fail to reject the null hypothesis. The z-table provides the cumulative probability or area under the standard normal curve to the left of a given z-score, which is used to calculate the p-value. The p-value represents the probability of obtaining a test statistic as extreme or more extreme than the one observed, given that the null hypothesis is true. If the p-value is less than the chosen significance level, the null hypothesis is rejected, indicating that the observed result is unlikely to have occurred by chance alone.
• Describe how the z-table is utilized when comparing two independent population proportions.
  • When comparing two independent population proportions, the z-table is used to calculate the test statistic and the corresponding p-value. The test statistic is a z-score that represents the difference between the two sample proportions, standardized by the standard error of the difference. The z-table is then used to find the cumulative probability or area under the standard normal curve to the left of the calculated z-score, which gives the p-value. The p-value is compared to the chosen significance level to determine if the difference between the two population proportions is statistically significant. If the p-value is less than the significance level, the null hypothesis (that the two population proportions are equal) is rejected, indicating that the observed difference is unlikely to have occurred by chance alone.
• Analyze how the properties of the standard normal distribution and the z-table are crucial for making inferences about population parameters in hypothesis testing and comparing two independent population proportions.
  • The standard normal distribution and the z-table are fundamental to hypothesis testing and comparing two independent population proportions because they allow researchers to quantify the likelihood of observed sample statistics occurring under the null hypothesis. The z-table provides the cumulative probability or area under the standard normal curve, which is used to calculate p-values and determine critical values for test statistics. This enables researchers to make informed decisions about whether to reject or fail to reject the null hypothesis based on the observed data. The standardized nature of the z-score and the z-table ensures that the same statistical procedures can be applied to any normal distribution, making it a versatile and essential tool for statistical inference. By understanding the properties of the standard normal distribution and the z-table, researchers can draw valid conclusions about population parameters and the statistical significance of their findings.

Related terms

Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1, which is used as a reference distribution for z-scores and hypothesis testing.

z-score: A standardized score that represents the number of standard deviations a data point is from the mean of a normal distribution.

Hypothesis Testing: A statistical method used to determine whether a claim about a population parameter is likely to be true or false based on sample data.