Question

The table below shows the critical reading scores for 14 students the first two times they took a standardized test. At α =0.01,

is there enough evidence to conclude that their scores improved the second time they took the​ test? Assume the samples are random and​ dependent, and the population is normally distributed. Complete parts​ (a) through​ (f).

Student	1	2	3	4	5	6	7	8	9	10	11	12	13	14
Score on first test	409	360	365	406	605	489	387	384	605	528	321	362	362	321
Score on second testScore on second test	418	437	442	454	536	533	386	520	673	581	331	390	392	342

a) Identify the claim and state H0 and Ha.

The claim is​ "The students' critical reading test scores ▼ (decreased, did not change, changed, improved) the second time they took the​ test.

Let μd be the hypothesized mean of the the​ students' first score minus their second score. State H0 and Ha.

Choose the correct answer below.

A. H0​: μ≥d

Ha​: μ<d

B. H0​: μ≤d

Ha​: μ>d

C. H0​: μ≥0

Ha​: μ<0

D.

H0​: μ≠0

Ha​:μ=0

E. H0​: μ≤0

Ha​:μ>0

F. H0​: μ=0

Ha​: μ≠0

​(b) Find the critical​ value(s) and identify the rejection​ region(s).

t 0= ​(Use a comma to separate answers as needed. Type an integer or a decimal. Round to three decimal places as​ needed.)

Identify the rejection​ region(s). Choose the correct answer below.

A. t<−2.650 or t>2.650

B. t<−2.650

C. t<−3.012 or t>3.012

D. t>3.012

​(c) Calculate d and sd.

d= ​(Type an integer or a decimal. Round to three decimal places as​ needed.)

Calculate sd.

sd=   Type an integer or a decimal. Round to three decimal places as​ needed.)

​(d) Use the​ t-test to find the standardized test statistic t.

t= ​(Type an integer or a decimal. Round to three decimal places as​ needed.)

​(e) Decide whether to reject or fail to reject the null hypothesis. Choose the correct answer below.

Fail to reject the null hypothesis.

Rejectthe null hypothesis.

​(f) Interpret the decision in the context of the original claim. Choose the correct answer below.

A. At the​ 1% significance​ level, there is not enough evidence that the​ students' critical reading scores improved the second time they took the test.

B. At the​ 1% significance​ level, there is enough evidence that the​ students' critical reading scores improved the second time they took the test.

C. The sample was not large enough to make a conclusion.

D. At the​ 1% significance​ level, there is evidence that the​ students' critical reading scores got worse the second time they took the test

Answer 1

For doing hypothesis testing first we need to find Mean of differences (d̄) and Standard deviation of the differences (s_d).

Following is the calculation showing how we find d̄ and s_d

Step 1: Set up null and alternative hypotheses.

H₀: μ_d> 0 (There is no difference in first and second test scores)
H_a: μ_d < 0 (The second test scores are higher than first test scores)

Step 2: Input α (level of significance of hypothesis test).

α = 0.01 (Note: α = level of significance of hypothesis test = probability of making Type I error.)

Step 3: Calculate Test Statistic.

d̅ (sample mean) = -37.929
s_d =  47.194
n (sample size) = 14

          (Note: From Step 1, we have H₀: μ_d = 0; therefore set μ_d = 0)

t = (-37.929 - 0)/(47.194/SQRT(14))

t = -3.0071

Step 4: Find Critical Value

Note: Since σ is unknown, t-distribution is used to find the critical value.
t_α is the t-score corresponding to the left-tailed area. Degrees of freedom = n-1 = 14 - 1 = 13

t_0.01 = -2.650 (Obtained using t distribution table)

Rejection Region H₀: t < -2.650

Step 5: Make Decision
Test Statistic = -3.007
Critical Values is -2.650
In this case, the test statistic is less than or equal to the critical value, so we reject the null hypothesis.

Answer a): Option C is correct

C. H₀​: μ≥0 H_a​: μ<0

Answer b)

Critical Value t₀ = -2.650

Rejection Region Option B is correct

t₀ < -2.650

Answer c)

d̅ = -37.929
s_d =  47.194

Answer d)

Test Statistics = -3.007

Answer e)

Reject the null hypothesis (Because Test Statistics < Critical Value)

Answer f) Option B is correct

At the​ 1% significance​ level, there is enough evidence that the​ students' critical reading scores improved the second time they took the test.