# Confidence interval for a population mean with a known and an unknown variance

When population variance is **known**, and the distribution is normal, then z-statistic is used for all sample sizes for calculating reliability factor.

When the population variance is **unknown**, and the distribution is normal, then t-statistic is used for small sample sizes. For large sample sizes, even z-statistic can also be used because t-distribution approaches normal distribution as the sample size increases. However, t-statistic would give a more accurate confidence interval. The confidence interval using t-statistic is always wider than the confidence interval of z-statistic.

When the population distribution is **non-normal**, and the sample size is small, then no statistic is available for computing reliability factor. However, when the sample size is greater than or equal to 30, then the distribution of sample statistic approaches normal even if the underlying population distribution is non-normal. Then, either of z-statistic or t-statistic can be used for an unknown variance, and z-statistic is used for a known variance.

Example 2: Calculating confidence interval for normal distribution |

The annual returns of a mutual fund are following a normal distribution. What is the 95 percent confidence interval for the population mean of annual returns if the sample mean is 8.50 percent and the sample size is 25? |

Example 3: Calculating confidence interval for nonnormal distribution |

Ross is considering an investment in a hedge fund. He wants to know the 95 percent confidence interval for the population mean of the annual returns of a hedge fund. The average annual return of the hedge fund is 12 percent. The population standard deviation of the returns is 20 percent. The returns of the hedge fund do not follow a normal distribution. What is the 99 percent confidence interval given the sample size is: |

Statistic to be used for computing reliability factors |
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Sample taken from |
Small sample size |
Large sample size |

Normal distribution with known variance |
z-statistic |
z-statistic |

Normal distribution with unknown variance |
t-statistic |
t-statistic or z-statistic |

Nonnormal distribution with known variance |
Not available |
z-statistic |

Nonnormal distribution with unknown variance |
Not available |
t-statistic or z-statistic |

**The impact of sample size on the width of confidence interval:** The larger is the sample size, the narrower is the width of the confidence interval. The width of the confidence interval depends on two factors: Reliability factor and standard error. The standard error decreases with an increase in sample size because it is equal to s/√n. The reliability factor also increases with increase in the sample size (n) because then the degrees of freedom increases and the reliability factor decreases with an increase in degrees of freedom.

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