Question

part 3: hypothesis test for a proportion (5 mins) innovate inc. claims that more than 25% of its enterprise clients upgrade to the premium tier within their first year. in a random sample of 200 enterprise clients, 60 of them upgraded. test this claim at a 1% level of significance (α = 0.01). 6. state the hypotheses: h0: ha: 7. calculate the sample proportion (p) and the test statistic (z - statistic):

Explanation:

Step1: Calculate sample proportion

The sample proportion $p'$ is calculated as the number of successes divided by the sample size. Here, the number of successes (clients who upgraded) is $n_1 = 60$ and the sample size $n=200$. So, $p'=\frac{60}{200}=0.3$.

Step2: State the null and alternative hypotheses

The null hypothesis $H_0$ is the status - quo or the claim we assume to be true initially. The alternative hypothesis $H_a$ is what we are trying to find evidence for. The company claims that more than 25% of its enterprise clients upgrade to the premium tier. So, $H_0:p\leq0.25$ and $H_a:p > 0.25$, where $p$ is the population proportion.

Step3: Calculate the test statistic

The formula for the z - statistic in a one - sample proportion test is $z=\frac{p' - p}{\sqrt{\frac{p(1 - p)}{n}}}$. Substitute $p' = 0.3$, $p = 0.25$, and $n = 200$ into the formula. First, calculate the denominator $\sqrt{\frac{p(1 - p)}{n}}=\sqrt{\frac{0.25\times(1 - 0.25)}{200}}=\sqrt{\frac{0.25\times0.75}{200}}=\sqrt{\frac{0.1875}{200}}\approx\sqrt{0.0009375}\approx0.0306$. Then, $z=\frac{0.3 - 0.25}{0.0306}=\frac{0.05}{0.0306}\approx1.63$.

Answer:

The sample proportion $p' = 0.3$. The null hypothesis $H_0:p\leq0.25$, the alternative hypothesis $H_a:p > 0.25$, and the test statistic $z\approx1.63$.