www-awa.aleks.com/alekscgi/x/lsLexe/1o_u-IgNsikr7j8P3jH-IBddblg73xc1DSqGhHBiU1CqDfFo3C4Mu91z42nFNviKSu_MZg99xeiTgk5IAqYNcekuMW554XnywRdEy0MmBZu9bOCNOuGy?1oBw7QYil
Conndance Intervals and Hypoihesh Testing
Confidence interval for the population mean: Use of the t distribution
Kennedee

Suppose that a random sample of 20 adults has a mean score of 64 on a standardized personality test, with a standard deviation of 6 . (A higher score indicates a more personable participant.) If we assume that scores on this test are normally distributed, find a confidence interval for the mean score of all takers of this test. Give the lower limit and upper limit of the confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)

Lower limit:
Upper limit:

Ask by West Guzman. in the United States

Mar 27,2025

Question

www-awa.aleks.com/alekscgi/x/lsLexe/1o_u-IgNsikr7j8P3jH-IBddblg73xc1DSqGhHBiU1CqDfFo3C4Mu91z42nFNviKSu_MZg99xeiTgk5IAqYNcekuMW554XnywRdEy0MmBZu9bOCNOuGy?1oBw7QYil
Conndance Intervals and Hypoihesh Testing
Confidence interval for the population mean: Use of the t distribution
Kennedee

Suppose that a random sample of 20 adults has a mean score of 64 on a standardized personality test, with a standard deviation of 6 . (A higher score indicates a more personable participant.) If we assume that scores on this test are normally distributed, find a confidence interval for the mean score of all takers of this test. Give the lower limit and upper limit of the confidence interval.

Carry your intermediate computations to at least three decimal places. Round your answers to one decimal place. (If necessary, consult a list of formulas.)

Lower limit:
Upper limit:

Ask by West Guzman. in the United States

Mar 27,2025

UpStudy AI · Accepted Answer

**Step 1. Identify the parameters**

- Sample size: $$ n = 20 $$
- Sample mean: $$ \bar{x} = 64 $$
- Sample standard deviation: $$ s = 6 $$
- Confidence level: $$ 99\% $$ (which gives a significance level $$ \alpha = 0.01 $$)
- Degrees of freedom: $$ df = n - 1 = 19 $$

---

**Step 2. Find the t critical value**

For a $$ 99\% $$ confidence interval, we need the critical value $$ t_{\alpha/2, \, df} $$ with $$ \alpha/2 = 0.005 $$ and $$ df = 19 $$. Consulting a t-distribution table, we find

$$
t_{0.005, \, 19} \approx 2.861.
$$

---

**Step 3. Compute the standard error of the mean**

The standard error (SE) is calculated as:

$$
SE = \frac{s}{\sqrt{n}} = \frac{6}{\sqrt{20}}.
$$

Calculating the square root:

$$
\sqrt{20} \approx 4.472,
$$

so

$$
SE \approx \frac{6}{4.472} \approx 1.3416.
$$

---

**Step 4. Calculate the margin of error**

The margin of error (ME) is:

$$
ME = t_{\alpha/2, \, df} \times SE \approx 2.861 \times 1.3416 \approx 3.840.
$$

---

**Step 5. Determine the confidence interval**

The $$ 99\% $$ confidence interval for the population mean is given by:

$$
\bar{x} \pm ME = 64 \pm 3.840.
$$

Thus, the limits are:

$$
\text{Lower limit} = 64 - 3.840 \approx 60.16,
$$
$$
\text{Upper limit} = 64 + 3.840 \approx 67.84.
$$

Rounding to one decimal place:

$$
\text{Lower limit} \approx 60.2,
$$
$$
\text{Upper limit} \approx 67.8.
$$

---

**Final Answer**

Lower limit: $$ 60.2 $$  
Upper limit: $$ 67.8 $$
Lower limit: 60.2 Upper limit: 67.8

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Statistics Questions