Question

An organization monitors many aspects of elementary and secondary education nationwide. Their 1995 numbers are often used as a baseline to assess changes. In 1995​, 42 % of students had not been absent from school even once during the previous school year. In the 1999 ​survey, responses from 7146 randomly selected students showed that this figure had slipped to 41 %. Officials would note any change in the rate of student attendance. Answer the questions below.

​(a) Write appropriate hypotheses.

Upper H 0 : The percentage of students in 1999 with perfect attendance the previous school year (1)_______

Upper H Subscript Upper A Baseline : The percentage of students in 1999 with perfect attendance the previous school year (2)_________

​(b) Check the necessary assumptions and conditions.

The independence assumption is (3)___________

The randomization condition is (4) __________

The​ 10% condition is (5)____________

The​ success/failure condition is (6) ​______________

(c) Perform the test and find the​ P-value. ​

P-value equals _____________ ​(Round to three decimal places as​ needed.)

​(d) State your conclusion. Consider probabilities less than 0.05 to be suitably unlikely.

A. We fail to reject the null hypothesis. There is not sufficient evidence to suggest that the percentage of students with perfect attendance in the previous school year has changed.

B. We fail to reject the null hypothesis. There is sufficient evidence to suggest that the percentage of students with perfect attendance in the previous school year has changed.

C. We can reject the null hypothesis. There is sufficient evidence to suggest that the percentage of students with perfect attendance in the previous school year has changed.

(1) is greater than 42%.

is different from 42%.

is less than 42%.

is equal to 42%.

(2) is less than 42%.

is different from 42%.

is greater than 42%.

is equal to 42%.

(3) satisfied. not satisfied.

(4) satisfied. not satisfied.

(5) not satisfied. satisfied.

(6) satisfied. not satisfied.

Answer 1

1) is equal to 42%. Or M = 42%

2) is less than 42% or M < 42%

3) satisfied        Reason =   sample are independent from each other, meaning that the measurements for each sample subject are in no way influenced by or related to the measurements of other subjects. In our case students are independent of each other

4) satisfied          Reason = we have selected 7146 students randomly in sample. Hence satisfied

5) satisfied       reason = since our sample size 7146 is quite large hence we can assume that sample is approx. 10% of the population.

The following information is provided:

The sample size is N = 7146,

the number of favorable cases is X = 0.41 * 7146 = 2929.86 ~ 2930,

and the sample proportion is

And

the significance level is α=0.05

(1) Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: p = 0.42

Ha: p < 0.42

(2) Rejection Region

Based on the information provided, the significance level is α=0.05, and

the critical value for a left-tailed test is z_c = -1.64

The rejection region for this left-tailed test is

R={z:z<−1.64}

(3) Test Statistics

The z-statistic is computed as follows:

(4) Decision about the null hypothesis

Since it is observed that

z = -1.709 &lt; z_c = -1.64

it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is p = 0.0437,

and since p = 0.0437<0.05,

it is concluded that the null hypothesis is rejected.

(5) Conclusion

It is concluded that the null hypothesis Ho is rejected.

Therefore, there is enough evidence to claim that the population proportion p is less than p_0​, at the α=0.05 significance level.

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