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Construct a [tex]$95\%$[/tex] confidence interval for a sampling distribution that has means of [tex]$\{4, 4, 4, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.8, 4.8, 4.9, 4.9, 4.9, 4.9, 5, 5, 5, 5, 5, 5, 5.1, 5.1, 5.1, 5.2, 5.2\}$[/tex]. The standard deviation of the population from which these samples were drawn was known to be [tex]$\sigma = 0.357$[/tex].

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Sure, I'll guide you through constructing a 95% confidence interval step-by-step for the given data.

### Step 1: Organize the given data

The given sample means are:
[tex]\[ 4, 4, 4, 4.2, 4.2, 4.3, 4.3, 4.3, 4.4, 4.4, 4.4, 4.4, 4.5, 4.5, 4.6, 4.7, 4.7, 4.7, 4.8, 4.8, 4.8, 4.9, 4.9, 4.9, 4.9, 5, 5, 5, 5, 5, 5, 5.1, 5.1, 5.1, 5.2, 5.2 \][/tex]

### Step 2: Calculate the sample mean ([tex]\(\bar{x}\)[/tex]) and sample size ([tex]\(n\)[/tex])

The sample mean is the average of the sample data:

[tex]\[ \bar{x} = \frac{4 + 4 + 4 + 4.2 + 4.2 + 4.3 + 4.3 + 4.3 + 4.4 + 4.4 + 4.4 + 4.4 + 4.5 + 4.5 + 4.6 + 4.7 + 4.7 + 4.7 + 4.8 + 4.8 + 4.8 + 4.9 + 4.9 + 4.9 + 4.9 + 5 + 5 + 5 + 5 + 5 + 5 + 5.1 + 5.1 + 5.1 + 5.2 + 5.2}{36} \][/tex]

The calculation gives us:
[tex]\[ \bar{x} = 4.675 \][/tex]

The sample size ([tex]\(n\)[/tex]) is the number of sample means provided:
[tex]\[ n = 36 \][/tex]

### Step 3: Calculate the standard error of the mean (SE)

The standard error (SE) is calculated using the population standard deviation ([tex]\(\sigma\)[/tex]) and the sample size ([tex]\(n\)[/tex]):

[tex]\[ SE = \frac{\sigma}{\sqrt{n}} \][/tex]

Given [tex]\(\sigma = 0.357\)[/tex]:

[tex]\[ SE = \frac{0.357}{\sqrt{36}} = \frac{0.357}{6} = 0.0595 \][/tex]

### Step 4: Determine the z-score for the confidence level

For a 95% confidence level, the z-score corresponding to the two-tailed value is typically 1.96. However, more precisely, using statistical tables or software, the z-score is approximately:

[tex]\[ z = 1.95996 \][/tex]

### Step 5: Calculate the margin of error (ME)

The margin of error (ME) is computed as follows:

[tex]\[ ME = z \times SE \][/tex]

Using the calculated values:

[tex]\[ ME = 1.95996 \times 0.0595 = 0.1166 \][/tex]

### Step 6: Construct the confidence interval

The 95% confidence interval is computed by adding and subtracting the margin of error from the sample mean:

[tex]\[ \text{Confidence Interval} = (\bar{x} - ME, \bar{x} + ME) \][/tex]

Substituting the calculated values:

[tex]\[ \text{Confidence Interval} = (4.675 - 0.1166, 4.675 + 0.1166) \][/tex]

[tex]\[ \text{Confidence Interval} = (4.5584, 4.7916) \][/tex]

### Conclusion

Therefore, the 95% confidence interval for the population mean, based on the given sample data, is approximately:
[tex]\[ (4.5584, 4.7916) \][/tex]
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