A random sample of 16 students selected from the student body of a large university had an average age of 25 years and a standar

Question

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A random sample of 16 students selected from the student body of a large university had an average age of 25 years and a standar

d deviation of 2 years. We want to determine if the average age of all the students at the university is significantly more than 24. Assume the distribution of the population of ages is normal. The p-value is between

Mathematics

1 answer:

PIT_PIT [3.9K] · Answer 1 · 2026-02-18T13:05:59+03:00

Answer:

The P-value ranges between 2.5% and 5% according to the t-table.

Step-by-step explanation:

A random sample of 16 students from a large university showed an average age of 25 years with a standard deviation of 2 years.

Let $\mu$ = true average age of all students at the university.

So, the Null Hypothesis, $H_0$ : $\mu \leq$ 24 years {indicating the average age is less than or equal to 24 years}

Alternate Hypothesis, $H_A$ : $\mu$ > 24 years {indicating the average age is significantly greater than 24 years}

Here we employ the One-sample t-test statistics as the population's standard deviation is unknown;

T.S. = $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, $\bar X$ = sample average age = 25 years

s = sample standard deviation = 2 years

n = sample size = 16

This gives us the test statistics = $\frac{25-24}{\frac{2}{\sqrt{16} } }$ ~ $t_1_5$

= 2

The value of the t-test statistics is 2.

Moreover, the P-value of the test-statistics can be found as follows;

P-value = P( $t_1_5$ > 2) = 0.034 {as per the t-table}

Thus, the P-value lies between 2.5% and 5% based on the t-table.