The proportion test compares the sample's proportion to the population's proportion or compares the sample's proportion to the proportion of another sample.

We use this test to check if the known proportion is statistically correct, based on the sample proportion and the sample size.

the null hypothesis assumes that the known proportion is correct. The statistical decision will be based on the difference between the know proportion and the sample proportion.

You may choose between the **binomial test**, which is more accurate, especially for the small sample size and the **normal approximation**.

We recommend using only the **binomial test**. If the tool won't be able to calculate the binomial distribution it will automatically calculate base on the normal approximation. depend on the sample size and how close is x to np. for a sample size smaller than 1000 any combination will be calculate based on the binomial distribution (when choosing the binomial test).

Example: It is known that the proportion of newborn males in the human race is 0.5122. The residence of Brobdingnag claims that in their country the proportion is smaller.

**Binomial distribution**- the probability for an event is identical- The population's proportion,
**p**, is known._{0}

Calculated based on a random sample from the entire population.

**p̂**- Sample proportion or**x**number of successes.

If you enter a value between 0 and 1 the tool will assume you entered**p̂**, if you fill number bigger than 1, the tool will assume you entered the number of successes**x**.**n**- Sample size

x distribution is binomial.

The binomial mean is μ = np, and the binomial standard deviation is: $$\sigma_x=\sqrt{np(1-p)}$$ The proportion p distributes with a mean of p

$$ p>p_0:\qquad\quad c=-\frac{1}{2n}\\ p\lt p_0:\qquad\quad c=-\frac{1}{2n}\\ |p-p_0|\lt \frac{1}{2n}:\;c=0 $$

When using the binomial distribution the test statistic is the number of successes: X.

Since the distribution is

The following will use the example of

x | p(X=x) | p(X≤x) | p(X≥x) |
---|---|---|---|

0 | 0.100112915 | 0.100112915 | 1 |

1 | 0.266967773 | 0.367080688 | 0.899887085 |

2 | 0.311462402 | 0.678543091 | 0.632919312 |

3 | 0.207641602 | 0.886184692 | 0.321456909 |

4 | 0.086517334 | 0.972702026 | 0.113815308 |

5 | 0.023071289 | 0.995773315 | 0.027297974 |

6 | 0.003845215 | 0.99961853 | 0.004226685 |

7 | 0.000366211 | 0.999984741 | 0.00038147 |

8 | 1.52588E-05 | 1 | 1.52588E-05 |

$$p-value=p(X\le x)=\sum_{i=0}^{x}\binom{n}{x}p^xq^{n-x}$$ Example: x=1.

Since x < np, x located on the left side of the distribution. (1<8*0.25)

p-value=p(X≤1)=0.367081.

Since x < np, x located on the left side of the distribution. (1<8*0.25)

p-value=p(X≤1)=0.367081.

$$p-value=p(X\ge x)=\sum_{i=x}^{n}\binom{n}{x}p^xq^{n-x}$$ Example: x=4.

Since x > np, x located in the right side of the distribution. (4>8*0.25)

p-value=p(X ≥ 4)=1-p(X ≤ (4-1))=0.113815.

Since x > np, x located in the right side of the distribution. (4>8*0.25)

p-value=p(X ≥ 4)=1-p(X ≤ (4-1))=0.113815.

Find the tail in one size based on x.

Find the**x'** on the opposite tail, with the greater density that is less or equal to the density of **x**.

For example, if**x** on the left tail: $$p-value=p(X\le x) + p(X\ge x')\\ p-value=\sum_{i=0}^{x}\binom{n}{x}p^xq^{n-x} + \sum_{i=x}^{n}\binom{n}{x}p^xq^{n-x}$$ Example: x=1.

On the left side: p(X=1)=0.266968

On the right side: p(X=3)=0.207642. p(x=2)=0.311462, so x'=3.

p-value = p(X≤1) + p(X ≥ 3) = 0.367081 + 0.321457 = 0.6885376.

Find the

For example, if

On the left side: p(X=1)=0.266968

On the right side: p(X=3)=0.207642. p(x=2)=0.311462, so x'=3.

p-value = p(X≤1) + p(X ≥ 3) = 0.367081 + 0.321457 = 0.6885376.

The tool calculates the **h effect size**.

$$\varphi(p)=2arcsine(\sqrt{p})\\ h=\varphi(p̂)-\varphi(P_0)$$ Cohen's interpretation for the h effect size:

Small effect - 0.2.

Medium effect - 0.5.

Large effect - 0.8.

We use this test to check if the proportion of group1 is the same as the proportion of group2.

The tool's null hypothesis assumes that **the known difference between the groups is zero **(using only the pooled variance).

Example: compares the proportion of good oranges between two fields, base on a sample from each group. H_{0} assumes the proportions are identical.

**Binomial distribution**- the probability for an event within each group is identical

Calculated based on a random sample from the entire population

**p̂**,_{1}**p̂**- Sample proportions or_{2}**x**number of successes._{1},x_{2}

If you enter a value between 0 and 1 the tool will assume you entered**p̂**, if you fill number bigger than 1, the tool will assume you entered the number of successes**x**.**n**- Sample sizes_{1}, n_{2}

X

The difference between P

When H

The tool doesn't calculate the unpooled variance.

When H

$$ Right-tailed\;or\;Two\;tailed:\; c=\frac{1}{2n_1}+\frac{1}{2n_2}\\ Left-tailed\;or\;Two\;tailed:\; c=-\frac{1}{2n_1}+\frac{1}{2n_2}\\ $$

The tool calculates the **h effect size**.

$$\varphi(p)=2arcsine(\sqrt{p})\\ h=\varphi(p̂_1)-\varphi(p̂_2)$$ Cohen's interpretation for the h effect size:

Small effect - 0.2.

Medium effect - 0.5.

Large effect - 0.8.