Question

An article in the San Jose Mercury News stated that students in the California state university system take 5.5 years, on average, to finish their undergraduate degrees. A freshman student believes that the mean time is less and conducts a survey of 44 students. The student obtains a sample mean of 4.1 with a sample standard deviation of 1. Is there sufficient evidence to support the student's claim at an significance level? What is the t value and p value and how do I find this on a ti84

Answer 1

a. No, it is not safe to assume
`\(n \leq 5\%\)` without information about the total student population. b. Yes, n = 36 is greater than 30, meeting the Central Limit Theorem criterion for normality.

a. Is it safe to assume that n <= 5% of all college students in the local area?

No, it is not safe to assume n <= 5% without knowing the total number of college students in the local area. The 5% guideline is typically used when sampling from a large population.

b. Yes. The central limit theorem suggests that for large enough sample sizes (typically
`\( n \geq 30 \))`, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution.

Now, to test the student's claim about the mean time to finish undergraduate degrees:

Given:

- Population mean
`(\( \mu \))` = 5.5 years

- Sample mean
`(\( \bar{x} \))` = 6.3 years

- Sample standard deviation
`(\( s \))` = 0.9

- Sample size n = 36

- Significance level
`(\( \alpha \))` = 0.05

Test hypothesis:

`\[ H_0: \mu = 5.5 \]\[ H_a: \mu > 5.5 \]`

Calculate the test statistic t:

`\[ t = \frac{(\bar{x} - \mu)}{(s/√(n))} \]`

Substitute the values:

`\[ t = ((6.3 - 5.5))/((0.9/√(36))) \]\[ t = (0.8)/(0.15) \]\[ t = 5.33 \]`

Now, find the p-value associated with t = 5.33 and
`\( df = n - 1 = 35 \)`. This can be done using statistical software or consulting a t-distribution table.

If the p-value is less than
`\( \alpha \)` (0.05), reject the null hypothesis.

Conclusion:

If the p-value is less than 0.05, there is sufficient evidence to support the student's claim that the mean time to finish undergraduate degrees is less than 5.5 years.

The complete question is:

An article in the San Jose Mercury News stated that students in the California state university system take 5.5 years, on average, to finish their undergraduate degrees. A freshman student believes that the mean time is less and conducts a survey of 36 students. The student obtains a sample mean of 6.3 with a sample standard deviation of 0.9. Is there sufficient evidence to support the student's claim at an alpha = 0.05 significance level?

Preliminary:

a. Is it safe to assume that n =< 5% of all college students in the local area?

No

Yes

b. is n >= 30

Yes

No