Basic Computation: Testing p A random sample of 60 binomials...

Basic Computation: Testing $p$ A random sample of 60 binomials trials resulted in 18 successes. Test the claim that the population proportion of successes exceeds $18 \% .$ Use a level of significance of 0.01. (a) Check Requirements Can a normal distribution be used for the $\hat{p}$ distribution? Explain. (b) State the hypotheses. (c) Compute $\hat{p}$ and the corresponding standardized sample test statistic. (d) Find the $P$-value of the test statistic. (e) Do you reject or fail to reject $H_{0}$ ? Explain. (f) Interpretation What do the results tell you?

Basic Computation: Testing $p$ A random sample of 60 binomials trials resulted in 18 successes. Test the claim that the population proportion of successes exceeds $18 \% .$ Use a level of significance of 0.01.
(a) Check Requirements Can a normal distribution be used for the $\hat{p}$ distribution? Explain.
(b) State the hypotheses.
(c) Compute $\hat{p}$ and the corresponding standardized sample test statistic.
(d) Find the $P$-value of the test statistic.
(e) Do you reject or fail to reject $H_{0}$ ? Explain.
(f) Interpretation What do the results tell you?

Key Concepts

p-value

The p-value is the probability, under the null hypothesis, of obtaining a test statistic at least as extreme as the one observed. It quantifies the evidence against the null hypothesis; a smaller p-value indicates stronger evidence to reject it. In hypothesis tests, the p-value is compared to the significance level to make a decision.

Standardized Test Statistic

The standardized test statistic, often referred to as the z-score in proportion tests, measures how many standard errors the observed sample proportion is from the null hypothesis value. This is calculated by subtracting the null hypothesis proportion from the sample proportion and then dividing by the standard error, which quantifies the natural variability in the sample proportion.

Significance Level

The significance level, often denoted by ?, is a predetermined threshold used to decide whether to reject the null hypothesis. It represents the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. In this problem, a significance level of 0.01 is set, meaning that there is a 1% risk of making a wrong rejection.

Normal Approximation for Proportions

The normal approximation to a binomial distribution is used when sample sizes are large enough to meet certain criteria (typically np and n(1-p) both greater than or equal to 10) so that the sampling distribution of the sample proportion is approximately normal. This allows us to use z-tests for inference on population proportions.

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. It is characterized by parameters n (the number of trials) and p (the probability of success in each trial), and is used when the outcomes can be classified into two categories, such as success and failure.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample to infer that a certain condition is true for a whole population. In the context of proportions, it involves setting up a null hypothesis (usually representing the status quo or no effect) and an alternative hypothesis (representing the effect or claim being tested) and then using sample data to determine which hypothesis is more plausible.

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