Respuesta :
Answer:
The 90% confidence interval for the mean time to graduate with a bachelor’s degree is (4.32, 4.84).
Yes, this confidence interval contradict the belief that it takes 4 years to complete a bachelor’s degree.
Step-by-step explanation:
The (1 - α)% confidence interval for population mean μ, when the population standard deviation is not known is:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
The information provided is:
[tex]\bar x=4.58\\s=1.10\\\alpha =0.10[/tex]
Compute the critical value of t for 90% confidence interval and (n - 1) degrees of freedom as follows:
[tex]t_{\alpha/2, (n-1)}=t_{0.10/2, (50-1)}=t_{0.05, 49}=1.671[/tex]
*Use a t-table for the probability.
Compute the 90% confidence interval for population mean μ as follows:
[tex]CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}[/tex]
[tex]=4.58\pm 1.671\times \frac{1.10}{\sqrt{50}}\\=4.58\pm 0.26\\=(4.32, 4.84)[/tex]
Thus, the 90% confidence interval for the mean time to graduate with a bachelor’s degree is (4.32, 4.84).
If a hypothesis test is conducted to determine whether it takes 4 years to complete a bachelor’s degree or not, the hypothesis will be:
Hₐ: The mean time it takes to complete a bachelor’s degree is 4 years, i.e. μ = 4.
Hₐ: The mean time it takes to complete a bachelor’s degree is different from 4 years, i.e. μ ≠ 4.
The decision rule based on a confidence interval will be:
Reject the null hypothesis if the null value is not included in the interval.
The 90% confidence interval for the mean time to graduate with a bachelor’s degree is (4.32 years, 4.84 years).
The null value, i.e. μ = 4 is not included in the interval.
The null hypothesis will be rejected at 10% level of significance.
Thus, it can be concluded that that time it takes to complete a bachelor’s degree is different from 4 years.
