Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as $$ 1,2,3 $$, and 4 , respectively. The results are listed below. Construct a confidence interval estimate of the mean. What is the practical use of the confidence interval? $$ 2,2,1,3,4,4,4,4,3,1 $$ b what is the connidence interval tor the population mean $$ \mu $$ ? $$ 2.1

Question

Samples of DNA are collected, and the four DNA bases of A, G, C, and T are coded as $$ 1,2,3 $$, and 4 , respectively. The results are listed below. Construct a confidence interval estimate of the mean. What is the practical use of the confidence interval? $$ 2,2,1,3,4,4,4,4,3,1 $$ b what is the connidence interval tor the population mean $$ \mu $$ ? $$ 2.1<\mu<3.5 $$ (Round to one decimal place as needed.) What is the practical use of the confidence interval? Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. A. The confidence interval can be used to estimate that $$ 90 \% $$ of all people have DNA bases between (Round to one decimal place as needed.) B. The confidence interval can be used to estimate that, with $$ 90 \% $$ confidence, the interval from 2.4 to 4.0 actually contains the true mean DNA base people. (Round to one decimal place as needed.) C. The given numbers are just substitutes for the four DNA base names, so the numbers do not measure or count anything, and they are at the nominal of measurement. The confidence interval has no practical use. ver

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We are given the following sample values of the coded DNA bases:

$$
2,\;2,\;1,\;3,\;4,\;4,\;4,\;4,\;3,\;1
$$

Because the population is assumed normally distributed (or the sample is small so we use the $$t$$‐distribution), we first compute the sample mean and sample standard deviation.

**Step 1. Compute the sample mean $$\bar{x}$$:**

$$
\bar{x}=\frac{2+2+1+3+4+4+4+4+3+1}{10}=\frac{28}{10}=2.8
$$

**Step 2. Compute the sample standard deviation $$s$$:**

We first compute the squared deviations from the mean:

$$
\begin{aligned}
(2-2.8)^2 &=0.64\
(2-2.8)^2 &=0.64\
(1-2.8)^2 &=3.24\
(3-2.8)^2 &=0.04\
(4-2.8)^2 &=1.44\
(4-2.8)^2 &=1.44\
(4-2.8)^2 &=1.44\
(4-2.8)^2 &=1.44\
(3-2.8)^2 &=0.04\
(1-2.8)^2 &=3.24\
\end{aligned}
$$

Summing these values we get:

$$
0.64+0.64+3.24+0.04+1.44+1.44+1.44+1.44+0.04+3.24=13.6
$$

Then the sample variance is:

$$
s^2=\frac{13.6}{10-1}=\frac{13.6}{9}\approx1.5111
$$

Thus, the sample standard deviation is:

$$
s\approx\sqrt{1.5111}\approx1.2294
$$

**Step 3. Determine the $$t$$-value for a 90\% confidence interval:**

With a sample size of $$n=10$$, the degrees of freedom are $$df=9$$.
For a 90\% confidence interval, the two-tailed critical value is $$t_{0.95,\,9}$$. This value is approximately:

$$
t_{0.95,\,9}\approx1.833
$$

**Step 4. Calculate the margin of error (MOE):**

$$
\text{MOE}=t\frac{s}{\sqrt{n}}=1.833\frac{1.2294}{\sqrt{10}}\approx1.833\frac{1.2294}{3.1623}\approx1.833\times0.389=0.713
$$

**Step 5. Construct the confidence interval:**

$$
\bar{x}\pm \text{MOE} = 2.8\pm 0.713
$$

This gives the confidence interval:

$$
(2.8-0.713,\;2.8+0.713)=(2.087,\;3.513)
$$

Rounded to one decimal place, the confidence interval is:

$$
2.1<\mu<3.5
$$

**Interpretation of the Confidence Interval:**

The given data code the DNA bases with the numbers $$1,2,3,4$$. However, these numbers are merely labels that represent the four bases (e.g., A, G, C, and T). Since the numbers do not have a natural ordering that conveys meaningful magnitudes (they are nominal in nature), calculating an arithmetic mean and its confidence interval does not provide meaningful biological information. Therefore, while the confidence interval was computed correctly in a statistical framework, its practical use in this context is limited.

The correct answer among the choices is:

**C.** The given numbers are just substitutes for the four DNA base names, so the numbers do not measure or count anything, and they are at the nominal level of measurement. The confidence interval has no practical use.
The confidence interval for the population mean $$ \mu $$ is $$ 2.1 < \mu < 3.5 $$. The confidence interval is useful for estimating that, with 90% confidence, the true mean DNA base value lies between 2.1 and 3.5.

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