Determine whether a normal sampling distribution can be...

Determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions $p_{1}$ and $p_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent. Claim: $p_{1} \neq p_{2} ; \alpha=0.01$ Sample statistics: $x_{1}=35, n_{1}=70$ and $x_{2}=36, n_{2}=60$

Determine whether a normal sampling distribution can be used. If it can be used, test the claim about the difference between two population proportions $p_{1}$ and $p_{2}$ at the level of significance $\alpha .$ Assume the samples are random and independent.
Claim: $p_{1} \neq p_{2} ; \alpha=0.01$
Sample statistics: $x_{1}=35, n_{1}=70$ and $x_{2}=36, n_{2}=60$

Key Concepts

Two-Sample Z-Test for Proportions

This hypothesis test is used when comparing two independent population proportions to evaluate if there is a significant difference between them. It involves calculating a test statistic based on the difference between sample proportions and the standard error of the difference. The test statistic is then compared to a critical value from the normal distribution to make inferences about the populations.

Pooled Proportion

When testing the hypothesis of equality between two population proportions, a pooled proportion is often used to estimate the common proportion. This aggregate measure is calculated by dividing the total number of successes from both samples by the total number of observations. It is then used to compute the standard error under the null hypothesis that the two proportions are equal.

Normal Sampling Distribution

This concept refers to the idea that the sampling distribution of a sample proportion can be approximated by a normal distribution when the sample size is sufficiently large. The normal approximation is predicated on the binomial distribution being well-behaved under conditions such as np and n(1-p) being greater than or equal to a threshold (often 5). This allows statisticians to use z-scores and standard normal tables for inference.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the underlying population distribution, the distribution of the sample means (or proportions) tends to be normal if the sample size is large enough. In the context of proportions, as long as the sample sizes meet the necessary conditions, the distribution of the sample proportion will approximate a normal distribution, facilitating the use of parametric tests.

Significance Level

The significance level, denoted by alpha (?), is a threshold that defines the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. In this context, an alpha level of 0.01 indicates a 1% risk of concluding that there is a difference between the two population proportions when there is none.

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