Complete the following steps to interpret a 1-sample Z-test. Key output includes the estimate of the mean, the confidence interval, the p-value, and several graphs.

First, consider the sample mean, and then examine the confidence interval.

The mean of the sample data is an estimate of the population mean. Because the mean is based on sample data and not on the entire population, it is unlikely that the sample mean equals the population mean. To better estimate the population mean, use the confidence interval.

The confidence interval provides a range of likely values for the population mean. For example, a 95% confidence level indicates that if you take 100 random samples from the population, you could expect approximately 95 of the samples to produce intervals that contain the population mean. The confidence interval helps you assess the practical significance of your results. Use your specialized knowledge to determine whether the confidence interval includes values that have practical significance for your situation. If the interval is too wide to be useful, consider increasing your sample size. For more information, go to Ways to get a more precise confidence interval.

N | Mean | StDev | SE Mean | 95% CI for μ |
---|---|---|---|---|

20 | 16.460 | 2.258 | 0.581 | (15.321, 17.599) |

In these results, the estimate of the population mean for fat percentage is 16.46%. You can be 95% confident that the population mean is between 15.321% and 17.599%.

To determine whether the difference between the population mean and the hypothesized mean is statistically significant, compare the p-value to the significance level. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference.

- P-value ≤ α: The difference between the means is statistically significant (Reject H
_{0}) - If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis. You can conclude that the difference between the population mean and the hypothesized mean is statistically significant. Use your specialized knowledge to determine whether the difference is practically significant. For more information, go to Statistical and practical significance.
- P-value > α: The difference between the means is not statistically significant (Fail to reject H
_{0}) - If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis. You do not have enough evidence to conclude that the difference between the population mean and the hypothesized mean is statistically significant. You should make sure that your test has enough power to detect a difference that is practically significant. For more information, go to Power and Sample Size for 1-Sample Z.

N | Mean | StDev | SE Mean | 95% CI for μ |
---|---|---|---|---|

20 | 16.460 | 2.258 | 0.581 | (15.321, 17.599) |

Null hypothesis | H₀: μ = 15 |
---|---|

Alternative hypothesis | H₁: μ ≠ 15 |

Z-Value | P-Value |
---|---|

2.51 | 0.012 |

In these results, the null hypothesis states that the mean fat percentage is 15%. Because the p-value is 0.012, which is less than the significance level of 0.05, the decision is to reject the null hypothesis and conclude that the population mean fat percentage is different from 15%.

Problems with your data, such as skewness and outliers, can adversely affect your results. Use graphs to look for skewness and to identify potential outliers.

When data are skewed, the majority of the data are located on the high or low side of the graph. Often, skewness is easiest to detect with a histogram or boxplot.

Data that are severely skewed can affect the validity of the p-value if your sample is small (less than 20 values). If your data are severely skewed and you have a small sample, consider increasing your sample size.

Outliers, which are data values that are far away from other data values, can strongly affect the results of your analysis. Often, outliers are easiest to identify on a boxplot.

Try to identify the cause of any outliers. Correct any data–entry errors or measurement errors. Consider removing data values for abnormal, one-time events (also called special causes). Then, repeat the analysis. For more information, go to Identifying outliers.