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3 years ago

Find the critical value zc necessary to form a confidence interval at the level of confidence shown below. c=0.80

Mathematics

1 answer:

Jobisdone [24]3 years ago

3 0

Answer:

1.28

Step-by-step explanation:

Given a confidence (c) = 0.80

α = 1 - c = 1 - 0.8 = 0.2

α/2 = 0.2/2 = 0.1

From the normal distribution table, The critical value necessary to form a confidence interval is gotten by getting the z value of α/2, which is the same as that of 0.4 (0.5 - 0.1). The z value of α/2 is:

$z_\frac{\alpha}{2} =z_{0.01}=1.28$

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3 years ago

A random sample of 50 recent college graduates results in a mean time to graduate of 4.58 years, with a standard deviation of 1.

Anna007 [38]

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:

$CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}$

The information provided is:

$\bar x=4.58\\s=1.10\\\alpha =0.10$

Compute the critical value of t for 90% confidence interval and (n - 1) degrees of freedom as follows:

$t_{\alpha/2, (n-1)}=t_{0.10/2, (50-1)}=t_{0.05, 49}=1.671$

*Use a t-table for the probability.

Compute the 90% confidence interval for population mean μ as follows:

$CI=\bar x\pm t_{\alpha/2, (n-1)}\times \frac{s}{\sqrt{n}}$

$=4.58\pm 1.671\times \frac{1.10}{\sqrt{50}}\\=4.58\pm 0.26\\=(4.32, 4.84)$

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.

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