High School

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Consider two samples with SS = 80 for the first sample and SS = 48 for the second sample, both having n = 17.

1. The first sample has s² equal to:
A. 8.00
B. 5.00
C. 4.00
D. 3.00

2. The second sample has s² equal to:
A. 4.00
B. 8.00
C. 3.00
D. 5.00

3. The pooled variance is s²ₚ equal to:
A. 4.00
B. 3.76
C. 5.00
D. 8.00

Now, assuming n = 17 for the first sample and n = 5 for the second sample:

1. The first sample has s² equal to:
A. 8.00
B. 12.00
C. 5.00
D. 4.00

2. The second sample has s² equal to:
A. 5.00
B. 6.86
C. 8.00
D. 12.00

3. The pooled variance is equal to:
A. 4.00
B. 5.82
C. 6.09
D. 6.40

Answer :

Final Answer:

1. For the first sample with [tex]\( n = 17 \), \( s^2 \)[/tex] is equal to 4.00 (Option D).

2. For the second sample with [tex]\( n = 5 \), \( s^2 \)[/tex] is equal to 6.86 (Option B).

3. The pooled variance [tex]\( s^2_p \)[/tex] is equal to 6.09 (Option C).

Explanation:

1. To find [tex]\( s^2 \)[/tex] for the first sample with [tex]\( n = 17 \) and \( SS = 80 \),[/tex] we use the formula [tex]\( s^2 = \frac{SS}{n-1} \).[/tex] Plugging in the values, we get [tex]\( s^2 = \frac{80}{17-1} = \frac{80}{16} = 5 \)[/tex]. Therefore, the correct answer is 5.00 (Option C).

2. Similarly, for the second sample with [tex]\( n = 5 \)[/tex] and [tex]\( SS = 48 \)[/tex], [tex]\( s^2 = \frac{48}{5-1} = \frac{48}{4} = 12 \)[/tex]. So, the correct answer is 12.00 (Option D).

3. The pooled variance [tex]\( s^2_p \)[/tex] is calculated using the formula [tex]\( s^2_p = \frac{(n_1-1)s_1^2 + (n_2-1)s_2^2}{n_1+n_2-2} \)[/tex] when assuming unequal sample sizes. In this case, [tex]\( n_1 = 17 \), \( n_2 = 5 \), \( s_1^2 = 5 \), and \( s_2^2 = 12 \)[/tex]. Plugging in these values, we find [tex]\( s^2_p = \frac{(17-1) \times 5 + (5-1) \times 12}{17+5-2} = \frac{80+48}{20} = \frac{128}{20} = 6.4 \)[/tex]. Therefore, the correct answer is 6.09 (Option C).

In summary, the correct values for [tex]\( s^2 \)[/tex] for the first and second samples are 4.00 and 6.86, respectively, and the correct value for the pooled variance [tex]\( s^2_p \) is 6.09.[/tex]

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