To compare the standard normal $z$-values with the corresponding $t$-values for a sample size of $n = 3$, we need to understand the relationship between $z$-values (used in large samples) and $t$-values (used in small samples where the population standard deviation is unknown).

For $n = 3$:

• Degrees of freedom ($df$) = $n - 1 = 3 - 1 = 2$.

We'll calculate the values for the given confidence levels using a $t$-distribution table and compare them to the standard normal $z$-values.

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(a) 98% Confidence Interval

• Standard normal $z$-value: 2.33 (already provided).
• $t$-value for $df = 2$ at 98% confidence: t = 4.303 \text{ (from t-table or calculator)}.

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(b) 90% Confidence Interval

• Standard normal $z$-value: 1.645 (already provided).
• $t$-value for $df = 2$ at 90% confidence: t = 2.920 \text{ (from t-table or calculator)}.

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(c) 95% Confidence Interval

• Standard normal $z$-value: 1.96 (already provided).
• $t$-value for $df = 2$ at 95% confidence: t = 4.303 \text{ (from t-table or calculator)}.

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Comparison of $z$-values and $t$-values

Confidence Level (%)	$z$-value	$t$-value (df = 2)
98%	2.33	4.303
90%	1.645	2.920
95%	1.96	4.303

Key Observations:

1. The $t$-values are consistently larger than the corresponding $z$-values because the $t$-distribution accounts for additional uncertainty in small samples.
2. The difference between $z$-values and $t$-values decreases as the sample size increases, converging as $n \to \infty$.

Would you like further details on how to interpret these results?

Related Questions:

1. Why does the $t$-distribution have heavier tails than the normal distribution?
2. How does increasing sample size impact $t$-values?
3. Can $z$-values ever be used in small-sample confidence intervals?
4. How do degrees of freedom affect the $t$-distribution?
5. What is the formula for calculating confidence intervals with $t$-values?

Tip: Always consider the sample size when choosing between $z$- and $t$-values for confidence intervals.