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NISA [10]
2 years ago
14

A sample of 50 observations is taken from an infinite population. The sampling distribution of : a.is approximately normal becau

se of the central limit theorem. b.cannot be determined. c.is approximately normal because is always approximately normally distributed. d.is approximately normal because the sample size is small in comparison to the population size.
Mathematics
1 answer:
Korvikt [17]2 years ago
3 0

Answer:

a.is approximately normal because of the central limit theorem.

Step-by-step explanation:

The central limit theorem states that if we have a population with mean μ and standard deviation σ and we take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

For any distribution if the number of samples n ≥ 30, the sample distribution will be approximately normal.

Since in our question, the sample of observations is 50, n = 50.

Since 50 > 30, then <u>our sample distribution will be approximately normal because of the central limit theorem.</u>

So, a is the answer.

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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 - <em>α</em>)% confidence interval for population mean <em>μ, </em>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 <em>t</em> 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 <em>t</em>-table for the probability.

Compute the 90% confidence interval for population mean <em>μ</em> 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:

<em>Hₐ</em>:<em> </em>The mean time it takes to complete a bachelor’s degree is 4 years, i.e. <em>μ </em>= 4.

<em>Hₐ</em>:<em> </em>The mean time it takes to complete a bachelor’s degree is different from 4 years, i.e. <em>μ </em>≠ 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. <em>μ </em>= 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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