Question

In order to conduct a hypothesis test for the population mean, a random sample of 24 observations is drawn from a normally distributed population. The resulting sample mean and sample standard deviation are calculated as 4.8 and 0.8, respectively. (You may find it useful to reference the appropriate table: z table or t table)

H₀: μ ≤ 4.5 against H_A: μ > 4.5

a-1. Calculate the value of the test statistic. (Round all intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

a-2. Find the p-value.

• p-value < 0.01

• 0.01 ≤ p-value < 0.025

• 0.025 ≤ p-value < 0.05

• 0.05 ≤ p-value < 0.10

• p-value ≥ 0.10

a-3. At the 5% significance level, what is the conclusion?

• Reject H₀ since the p-value is less than α.

• Reject H₀ since the p-value is greater than α.

• Do not reject H₀ since the p-value is less than α.

• Do not reject H₀ since the p-value is greater than α.

a-4. Interpret the results at αα = 0.05.

• We conclude that the population mean is greater than 4.5.

• We cannot conclude that the population mean is greater than 4.5.

• We conclude that the population mean differs from 4.5.

• We cannot conclude that the population mean differs from 4.5.

H₀: μ = 4.5 against H_A: μ ≠ 4.5

b-1. Calculate the value of the test statistic. (Round all intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

b-2. Find the p-value.

• p-value < 0.01

• 0.01 ≤ p-value < 0.025

• 0.025 ≤ p-value < 0.05

• 0.05 ≤ p-value < 0.10

• p-value ≥ 0.10

b-3. At the 5% significance level, what is the conclusion?

• Reject H₀ since the p-value is less than α.

• Reject H₀ since the p-value is greater than α.

• Do not reject H₀ since the p-value is less than α.

• Do not reject H₀ since the p-value is greater than α.

b-4. Interpret the results at αα = 0.05.

• We conclude that the population mean is greater than 4.5.

• We cannot conclude that the population mean is greater than 4.5.

• We conclude that the population mean differs from 4.5.

• We cannot conclude that the population mean differs from 4.5.

Answer 1

a-1)
Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (4.8 - 4.5)/(0.8/sqrt(24))
t = 1.837

a-2)
p-value = P(t > 1.837)
0.025 ≤ p-value < 0.05

a-3.
Reject H0 since the p-value is less than α.

a-4)
We conclude that the population mean is greater than 4.5

b-1)
Test statistic,
t = (xbar - mu)/(s/sqrt(n))
t = (4.8 - 4.5)/(0.8/sqrt(24))
t = 1.837

b-2)
p-value = 2*P(t > 1.837)
0.05 ≤ p-value < 0.10

b-3)
Do not reject H0 since the p-value is less than α.

b-4)
We cannot conclude that the population mean differs from 4.5.