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3. The mean annual tuition and fees in the year 2013-2014 academic year for a s...

Nov 13, 2024

Solution by Steps

step 1

The parameter being tested here is the mean tuition fees for private institutions in California

step 2

The null hypothesis (

H_0

) is that the mean tuition fees is equal to

36,000

, and the alternative hypothesis (

H_a

) is that the mean tuition fees differs from

36,000

. Thus, we have:

H_0: \mu = 36000

H_a: \mu \neq 36000

step 3

This is a two-tailed test because we are checking for a difference in either direction (greater than or less than

36,000

)

step 4

The t-distribution should be used here because the sample size is small (n=14) and the population standard deviation is unknown

step 5

To compute the test statistic, we use the formula:

t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

, where

\bar{x} = 37900

\mu_0 = 36000

s = 7200

, and

n = 14

. Calculating gives:

t = \frac{37900 - 36000}{7200 / \sqrt{14}} \approx 1.25

step 6

To analyze the sample data using the P-value method, we find the P-value corresponding to the calculated t-statistic. For

t = 1.25

with

df = n - 1 = 13

, the P-value is approximately

0.12

. Since this is a two-tailed test, we double the P-value:

P \approx 0.24

step 7

The critical value for a two-tailed test at

\alpha = 0.01

with

df = 13

is approximately

\pm 3.23

. Since

1.25

is within the range of

-3.23

3.23

, we fail to reject the null hypothesis

step 8

The conclusion is that there is not enough evidence to conclude that the mean tuition fees for private institutions in California differs from

36,000

at the

0.01

significance level

Answer

There is not enough evidence to conclude that the mean tuition fees for private institutions in California differs from

36,000

at the

0.01

significance level.

Key Concept

Hypothesis testing involves comparing a sample statistic to a population parameter to determine if there is enough evidence to reject a null hypothesis.

Explanation

In this case, we tested whether the mean tuition fees differ from a specified value using a two-tailed t-test, and the results indicated insufficient evidence to reject the null hypothesis.