Unknown standard deviation

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When entering raw data, the tool will run the Shapiro-Wilk normality test, calculate outliers and generate R code, as part of the test calculation.

validation message

Target: To check if the assumed variance ()σ^{2}_{0}) is statistically correct, based on a sample variance S^{2}.

the default is right tailed test, usually you worried only if the standard deviation is bigger then the expected one.

1. Right tailed example:

A factory uses machine to create screws. the standard deviation of the screws diameter must not exceed 0.1 mm.

2. Two tailed test example:

A factory uses a machine to create screws. recently the engineers upgraded a big module in the machine.

The quality assurance are interested to know if there was a change in the standard deviation of the screws diameter.

If the standard deviation is bigger the engineer must fix it to meet the expected quality.

if the standard deviation is smaller, sales may consider to sell screws with better standard, and higher price. Example1: A farmer calculated last year the variance of the apples' weight in his apple orchard σ^{2}_{0} equals 2 kg^{2}, based on big sample

Current year he checked a small sample of apples and the sample variance S^{2} equals 3 kg^{2}

Was the variance of the apple's weight changed this year?

The farmer will run the two-tailed test.

The farmer consulted with a statistician and told him that he only worry if the variance will be greater, smaller variance is not a concern.

The statistician recommend him to run the right-tailed test and check the H_{1}: &σ^{2}>2_{kg}

Hypotheses

H_{0}: σ^{2} ≥ = ≤ σ^{2}_{0}

H_{1}: σ^{2} < ≠ > σ^{2}_{0}

Test statistic

χ² distribution

Normal distribution | |

Independent observations | |

The population expected variance is known |

Sample standard deviation | |

Sample size |

The following R code should produce the same results:

The R code doesn't exclude outliers.__Examples__

1. Two-tailed test

A farmer calculated last year the average of the apples' weight in his apple orchard μ_{0} equals 17 kg, based on the entire population.

The current year he checked a small sample of apples and the sample average x̄ equals 18 kg

Has the average of the apple's weight changed this year?

The farmer calculates the sample standard deviation of the apple's weight.

2. Left-tailed test.

In the same example as above, the farmer only cares to know if the entire average is lesser this year.