Question

1. a teacher claims the average number of hours students sleep on school nights is 7.5 hours. a group of seniors believes the true average is less than 7.5 (because... homework, sports, and tiktok). ( h_0: mu = 7.5 ) ( h_a: mu < 7.5 ) evidence: a random sample of 50 students sample mean ( \bar{x} = 6.8 ) hours and standard deviation ( s = 1.9 ) hours unusualness: ( t* = square ) (three decimal places) chances: p - value = ( square ) (three decimal places)

Explanation:

Step1: Calculate test statistic

The formula for t-statistic is $t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}}$. Substitute $\bar{x}=6.8$, $\mu_0=7.5$, $s=1.9$, $n=50$:
$t = \frac{6.8 - 7.5}{1.9/\sqrt{50}} = \frac{-0.7}{1.9/7.0711} \approx \frac{-0.7}{0.2687} \approx -2.605$

Step2: Find p-value

Degrees of freedom $df = n-1 = 49$. For $t=-2.605$ (left-tailed test), use t-distribution table or calculator: p-value ≈ 0.006

Answer:

t = -2.605
p-value = 0.006