We have given the number of successes and the sample size...

We have given the number of successes and the sample size for a simple random sample from a population. In each case, do the following. a. Determine the sample proportion. b. Decide whether using the one-proportion z-test is appropriate. c. If appropriate, use the one-proportion z-test to perform the specified hypothesis test. $$x=3, n=100, H_{0}: p=0.04, H_{\mathrm{a}}: p \neq 0.04, \alpha=0.10$$

We have given the number of successes and the sample size for a simple random sample from
a population. In each case, do the following.
a. Determine the sample proportion.
b. Decide whether using the one-proportion z-test is appropriate.
c. If appropriate, use the one-proportion z-test to perform the specified hypothesis test.
$$x=3, n=100, H_{0}: p=0.04, H_{\mathrm{a}}: p \neq 0.04, \alpha=0.10$$

Key Concepts

Sample Proportion

The sample proportion is a statistic that represents the ratio of successes to the total sample size. It is calculated by dividing the number of observed successes by the total number of trials, providing an estimate of the population proportion.

One-Proportion Z-Test

The one-proportion z-test is a hypothesis test used to determine whether an observed sample proportion significantly differs from a claimed or hypothesized population proportion. It involves comparing the sample proportion to the population proportion under the assumption that the sampling distribution of the proportion is approximately normal.

Normal Approximation Conditions

For the one-proportion z-test to be valid, the sample size must be sufficiently large so that both np and n(1-p) are greater than or equal to a minimum threshold (commonly 5 or 10). This condition ensures that the binomial distribution can be approximated well by a normal distribution.

Hypothesis Testing Framework

In a hypothesis test, the null hypothesis represents a baseline or claim about a population parameter, while the alternative hypothesis represents a deviation from that claim. A significance level (alpha) is chosen to determine the threshold for rejection of the null hypothesis, and the test statistic (in this case, a z-score) is calculated to assess whether the sample data provide sufficient evidence against the null hypothesis.

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