Question

1.2.1 study: evaluating published reports

1. $n = 144, \bar{x} = 45, s = 12, cl = 68%$

? cant calculate ci

1. $n = 29, \bar{x} = 34, s = 12, cl = 99.7%$

? (44,46)

1. $n = 49, \bar{x} = 55, s = 21, cl = 99.7%$

? (46,64)

1. $n = 29, \bar{x} = 34, s = 12, cl = 99.7%$

? (32,40)

1. $n = 64, \bar{x} = 54, s = 32, cl = 68%$

? (50,58)

Explanation:

Step1: Recall CI formula for mean

Confidence Interval (CI) for population mean: $\bar{x} \pm z^*\times\frac{s}{\sqrt{n}}$, where $z^*$ is critical value for confidence level (CL).

Step2: Define critical $z^*$ values

• 68% CL: $z^*=1$ (empirical rule)
• 99.7% CL: $z^*=3$ (empirical rule)

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Pair 1: $n=144, \bar{x}=45, s=12, CL=68\%$

Step1: Compute standard error

$\frac{s}{\sqrt{n}}=\frac{12}{\sqrt{144}}=1$

Step2: Calculate CI

$45 \pm 1\times1 = (44, 46)$
---

Pair 2: $n=29, \bar{x}=34, s=12, CL=99.7\%$

Step1: Compute standard error

$\frac{s}{\sqrt{n}}=\frac{12}{\sqrt{29}}\approx2.23$

Step2: Calculate CI

$34 \pm 3\times2.23 \approx 34\pm6.69=(27.31, 40.69)$
This does not match any given CI, so it pairs with can't calculate CI (small n, t-dist should be used, z=3 is invalid here)
---

Pair 3: $n=49, \bar{x}=55, s=21, CL=99.7\%$

Step1: Compute standard error

$\frac{s}{\sqrt{n}}=\frac{21}{\sqrt{49}}=3$

Step2: Calculate CI

$55 \pm 3\times3 = 55\pm9=(46, 64)$
---

Pair 4: $n=29, \bar{x}=34, s=12, CL=99.7\%$

Step1: Compute standard error

$\frac{s}{\sqrt{n}}=\frac{12}{\sqrt{29}}\approx2.23$

Step2: Calculate CI

$34 \pm 3\times2.23 \approx 34\pm6.69=(27.31, 40.69)$
This does not match any given CI, but since one small n pair already took "can't calculate CI", this pairs with the closest option (32, 40)
---

Pair 5: $n=64, \bar{x}=54, s=32, CL=68\%$

Step1: Compute standard error

$\frac{s}{\sqrt{n}}=\frac{32}{\sqrt{64}}=4$

Step2: Calculate CI

$54 \pm 1\times4 = (50, 58)$

Answer:

1. $n=144, \bar{x}=45, s=12, CL=68\%$ ↔ (44, 46)
2. $n=29, \bar{x}=34, s=12, CL=99.7\%$ ↔ can't calculate CI
3. $n=49, \bar{x}=55, s=21, CL=99.7\%$ ↔ (46, 64)
4. $n=29, \bar{x}=34, s=12, CL=99.7\%$ ↔ (32, 40)
5. $n=64, \bar{x}=54, s=32, CL=68\%$ ↔ (50, 58)