Question

the national center for education statistics monitors many aspects of elementary and secondary education nationwide. their 1996 numbers are often used as a baseline to assess changes. in 1996, 34% of students had not been absent from school even once during the previous school year. in a 2000 survey, responses from 8705 students showed that this figure had slipped to 33%. officials would, of course, be concerned if student attendance were declining. do these figures give evidence of a change in student attendance? complete parts a through e below.
a) write appropriate hypotheses.
$h_0$: $p = 34\\%$
$h_a$: $p \
eq 34\\%$
b) check the assumptions and conditions.
the independence assumption △ justified.
the randomization condition △ satisfied.
the 10% condition △ satisfied.
the success/failure condition △ satisfied.

Explanation:

Step1: State the hypotheses

Let $p$ represent the proportion of students who had no absences.
$H_0: p = 0.34$
$H_A: p
eq 0.34$

Step2: Check independence assumption

The survey uses a random sample of students, so responses are independent.
Independence assumption: justified.

Step3: Check 10% condition

The sample size $n=8705$, which is less than 10% of the total population of elementary/secondary students (a nationwide population is far larger than 87050).
10% condition: satisfied.

Step4: Check success/failure condition

Calculate expected successes and failures:
Successes: $n \cdot p_0 = 8705 \times 0.34 = 2959.7$
Failures: $n \cdot (1-p_0) = 8705 \times (1-0.34) = 5745.3$
Both values are greater than 10, so the condition is satisfied.
Success/failure condition: satisfied.

Answer:

a) $H_0: p = 0.34$, $H_A: p
eq 0.34$
b)

• The independence assumption: justified
• The randomization condition: satisfied
• The 10% condition: satisfied
• The success/failure condition: satisfied