Write the null and alternative hypotheses for each of the following situations:

APioneer Communications advertised the following for one of their fiber optic internet packages: download speeds up to 10 Mbps. Suppose a competitor decides to test this claim and see if the average speed is actually 10 Mbps or if it is actually less than that.

BAccording to WorkStream.us, as of 2023, the median hourly wage in Kansas was $27.05, while the mean hourly wage was $19.44. Has the mean hourly wage changed since then?

CIn a population of fish, approximately 42 percent are female. A test is conducted to see if, in fact, the proportion of female fish is more.

DNationwide, the average length of time a dog stays at a shelter is 27 days. How could we do a hypothesis test to see if this appears to be true at Finney County Humane Society?

In hypothesis testing, we use two mutually exclusive statements about a population parameter (like the mean $\mu$ or the proportion $p$). The null hypothesis ($H_0$) represents the status quo or the claim to be tested, while the alternative hypothesis ($H_a$) represents the researcher's claim or the effect we are trying to detect. We designate a test as left-tailed if we are testing for a value "less than", right-tailed if "more than", and two-tailed if we are testing for any "difference" or "change".

Formulating hypotheses for part (a)

For part (a), the claim is that the average speed is $10$ Mbps. The competitor wants to test if the speed is actually less than that.

The null hypothesis is the baseline: $H_0: \mu = 10$ Mbps.
The alternative hypothesis is the suspicion: $H_a: \mu < 10$ Mbps.

This is a left-tailed test.

Formulating hypotheses for part (b)

For part (b), we are checking if the mean wage of $\$19.44$ has changed.

The null hypothesis is no change: $H_0: \mu = 19.44$.
The alternative hypothesis is that there is a change (it is not equal to $19.44$): $H_a: \mu eq 19.44$.

This is a two-tailed test.

Formulating hypotheses for part (c)

For part (c), we compare the sample to the known proportion of $42\%$ (or $0.42$) and want to see if the true proportion is more.

The null hypothesis is the baseline proportion: $H_0: p = 0.42$.
The alternative hypothesis is our research interest: $H_a: p > 0.42$.

This is a right-tailed test.

Formulating hypotheses for part (d)

For part (d), we are testing if the shelter's average of $27$ days differs from the national average of $27$ days.

The null hypothesis is that the shelter follows the national trend: $H_0: \mu = 27$.
The alternative hypothesis is that the shelter's average is different: $H_a: \mu eq 27$.

This is a two-tailed test.

Examples

Hypothesis testing is widely used in real-world decision-making. For example, if a pharmaceutical company develops a new drug, they use a null hypothesis ($H_0$) that the drug has no effect and an alternative hypothesis ($H_a$) that the drug effectively treats a condition. By calculating the probability of observing their data under $H_0$, researchers can make evidence-based decisions about whether to approve the new medication. This rigorous framework helps separate genuine effects from random chance.