At Least As Extreme A random sample of 860 births in New York State included 426 boys, and that sample is to be used for a test of the common belief that the proportion of male births in the population is equal to 0.512.

b. For random samples of size 860, what sample proportions of male births are at least as extreme as the sample proportion of 426/860?

검증된 단계별 안내

Calculate the sample proportion of male births using the formula:

\frac{426}{860}

. This will give the observed sample proportion.

Determine the null hypothesis proportion, which is given as 0.512. This represents the assumed proportion of male births in the population under the null hypothesis.

Compute the standard error (SE) of the sample proportion using the formula:

\sqrt{\frac{p (1 - p)}{n}}

, where

p

is the null hypothesis proportion (0.512) and

n

is the sample size (860).

Calculate the z-score for the observed sample proportion using the formula:

p

_obs-pSE, where

p

_obs is the observed sample proportion,

p

is the null hypothesis proportion, and

SE

is the standard error.

Find the z-scores that correspond to sample proportions at least as extreme as the observed proportion. This involves considering both tails of the distribution (greater than the observed proportion and less than the symmetric lower proportion). Use a z-table or statistical software to determine the corresponding probabilities.

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주요 개념

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Sample Proportion

The sample proportion is the ratio of the number of successes (in this case, male births) to the total number of observations in the sample. It is calculated by dividing the number of boys (426) by the total number of births (860), yielding a sample proportion of 0.495. This value is crucial for hypothesis testing as it serves as the basis for comparing against the hypothesized population proportion.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine whether there is enough evidence in a sample to infer that a certain condition holds for the entire population. In this context, the null hypothesis states that the proportion of male births is equal to 0.512. The test will evaluate if the observed sample proportion is significantly different from this hypothesized value, using a significance level to make a decision.

At Least As Extreme

The phrase 'at least as extreme' refers to the values of the sample proportion that are as far away from the hypothesized population proportion as the observed sample proportion. This concept is used to determine the critical values for the hypothesis test, which helps in assessing whether the observed sample proportion is statistically significant compared to the expected proportion under the null hypothesis.