Question

1. liam and zoe are both conducting a hypothesis test about teen habits. they have their own hypotheses and sample statistics. we want to determine which test has stronger evidence against their null (in favor of their alternative).

liam’s hypotheses:
$h_0: p = 0.60$
$h_a: p > 0.60$
liam’s evidence: 65% of 250 teens say they spend more than 2 hours per day on tiktok.

zoe’s hypotheses:
$h_0: p = 0.40$
$h_a: p < 0.40$
zoe’s evidence: 35% of 400 teens say they play video games more than 1 hour per day.

calculate the test statistic for both. (type everything into desmos at once and round your answers to the 3rd decimal place!)

liam’s t.s. =
zoe’s t.s. =

who has stronger evidence, liam or zoe?

Explanation:

Step1: Calculate Liam's sample proportion

$\hat{p}_L = \frac{60}{200} = 0.30$

Step2: Compute Liam's z-statistic

$z_L = \frac{\hat{p}_L - p_0}{\sqrt{\frac{p_0(1-p_0)}{n_L}}} = \frac{0.30 - 0.03}{\sqrt{\frac{0.03(1-0.03)}{200}}} = \frac{0.27}{\sqrt{\frac{0.0291}{200}}} = \frac{0.27}{\sqrt{0.0001455}} \approx \frac{0.27}{0.01206} \approx 22.39$

Step3: Calculate Zoe's sample proportion

$\hat{p}_Z = \frac{35}{600} \approx 0.0583$

Step4: Compute Zoe's z-statistic

$z_Z = \frac{\hat{p}_Z - p_0}{\sqrt{\frac{p_0(1-p_0)}{n_Z}}} = \frac{0.0583 - 0.03}{\sqrt{\frac{0.03(1-0.03)}{600}}} = \frac{0.0283}{\sqrt{\frac{0.0291}{600}}} = \frac{0.0283}{\sqrt{0.0000485}} \approx \frac{0.0283}{0.00696} \approx 4.07$

Step5: Compare z-statistics

Liam's z-statistic (22.39) > Zoe's z-statistic (4.07)

Answer:

Liam's z = 22.4
Zoe's z = 4.1
Stronger evidence: Liam