A friend is tested to see whether he can tell bottled water...

A friend is tested to see whether he can tell bottled water from tap water. There are 30 trials (half with bottled water and half with tap water), and he gets 18 right. a. Pick the correct null hypothesis: i. $\hat{p}=0.50$ ii. $\hat{p}=0.60$ iii. $p=0.50$ iv. $p=0.60$ b. Pick the correct alternative hypothesis: i. $\hat{p} \neq 0.50$ ii. $\hat{p}=0.875$ iii. $p>0.50$ iv. $p \neq 0.875$

A friend is tested to see whether he can tell bottled water from tap water. There are 30 trials (half with bottled water and half with tap water), and he gets 18 right.
a. Pick the correct null hypothesis:
i. $\hat{p}=0.50$
ii. $\hat{p}=0.60$
iii. $p=0.50$
iv. $p=0.60$
b. Pick the correct alternative hypothesis:
i. $\hat{p} \neq 0.50$
ii. $\hat{p}=0.875$
iii. $p>0.50$
iv. $p \neq 0.875$

Key Concepts

Null Hypothesis

The null hypothesis is a statement about the population parameter that indicates there is no effect or no difference from a specified value. It is the hypothesis that any observed difference is due to random chance. In hypothesis testing, it typically assumes that a particular value (such as 0.50 for a proportion) is true, and it is used as the baseline against which the alternative hypothesis is tested.

Alternative Hypothesis

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that the true population parameter is different from the stated value in the null hypothesis. It reflects the research question or the suspected effect or difference, and it can be one-sided or two-sided, depending on whether the interest lies in a difference in one direction only or in both directions.

Population Proportion

The population proportion is a parameter that represents the proportion of the entire population that has a particular characteristic. In the context of hypothesis testing, it is compared against the observed sample proportion to determine whether the sampling outcome is consistent with the hypothesis that the population proportion equals a specific value.

Hypothesis Testing for Proportions

Hypothesis testing for proportions is a statistical method used to decide whether a sample proportion significantly differs from a claimed population proportion. It involves comparing the observed sample data to what would be expected if the null hypothesis were true, and it employs probability models like the binomial or normal approximation depending on sample size.

Binomial Distribution

The binomial distribution is a probability distribution used to model the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes and the probability of success is constant. This distribution is particularly relevant in hypothesis testing for proportions when the data come from a series of yes/no experiments or trials.

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