A common way to calculate confidence limits is with a table of cumulative values for a "standard" Normal distribution. A standard Normal distribution has a mean of 0 and a standard deviation of 1. Most statistics books or books of numerical values will have a table of standard Normal figures. It is important to work with either a "two-tailed" table or double your answers from a "one-tail" table. The "tail" refers to the curve going in both directions from the mean in the center.

A portion of a two-tailed standard Normal table is given in Table 2-6. Look in this table for the "y" value. This is the displacement from the mean along the horizontal axis. Look at y = 1, one standard deviation from the mean. You will see an entry in the cumulative column of 0.6826. This means that the "area under the curve" from ±1 s is 0.6826 of all the area. The confidence of a value falling around the mean, ±1s , is 0.6826, commonly truncated to 68.3%.

"y" Value |
Probability |

0.1 |
0.0796 |

0.2 |
0.1586 |

0.3 |
0.2358 |

0.4 |
0.3108 |

0.5 |
0.4514 |

0.6 |
0.5160 |

0.7 |
0.5762 |

0.8 |
0.6318 |

1.0 |
0.6826 |

1.1 |
0.7286 |

1.2 |
0.7698 |

1.3 |
0.8064 |

1.4 |
0.8384 |

1.5 |
0.8664 |

1.6 |
0.8904 |

1.7 |
0.9108 |

1.8 |
0.9282 |

1.9 |
0.9426 |

2.0 |
0.9544 |

2.1 |
0.9643 |

2.2 |
0.9722 |

2.3 |
0.9786 |

2.4 |
0.9836 |

2.5 |
0.9876 |

2.6 |
0.9907 |

2.7 |
0.9931 |

2.8 |
0.9949 |

2.9 |
0.9963 |

3.0 |
0.9974 |

For p(-y < Xi < y) where Xi is a standard normal random variable of mean 0 and standard deviation of 1._

For nonstandard Normal distributions, look up y = a/s , where "a" is the value from a nonstandard distribution with mean = 0 and s is the standard deviation of that nonstandard Normal distribution._

If the mean of the nonstandard Normal distribution is not equal to 0, then "a" is adjusted to "a = b - p," where "b" is the value from the nonstandard Normal distribution with mean p: y = (b - p)/s ._

t23]For a normal curve, the slope changes such that the curvature goes from concave to convex at exactly ±1 s from the mean. This curvature change will show up as an inflection on the cumulative probability curve.

Team LiB

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