Question

Math

Posted 4 months ago

```
Esteban suspects that right-handed participants have an unfair advantage on a memory test because the input device uses the right hand. He selects a random sample of existing test data and summarizes it in the following table.
\begin{tabular}{lll}
\multicolumn{3}{c}{ Scores on memory test (points) } \\
& Left-handed & Right-handed \\
\hline Sample mean & $\bar{x}_{\mathrm{L}}=115.3$ & $\bar{x}_{\mathrm{R}}=116.8$ \\
Sample standard deviation & $s_{\mathrm{L}}=26.2$ & $s_{\mathrm{R}}=15.8$ \\
Sample size & $n_{\mathrm{L}}=9$ & $n_{\mathrm{R}}=24$
\end{tabular}
Assume that the conditions for inference have been met.
Let $\mu_{\mathrm{R}}-\mu_{\mathrm{L}}$ be the difference in mean score of the right- and left-handed participants.
Which of the following is a $90 \%$ confidence interval for $\mu_{\mathrm{R}}-\mu_{\mathrm{L}}$ ?
Use a calculator with statistical capabilities to calculate the interval.
Choose 1 answer:
(A) $(-27.84,30.84)$
(B) $(-19.17,22.17)$
(C) $(-15.33,18.33)$
(D) $(-13.81,16.81)$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

Substitute the given values into the formula: $SE = \sqrt{\frac{26.2^2}{9} + \frac{15.8^2}{24}}$

step 3

Calculate the value of SE: $SE = \sqrt{\frac{686.44}{9} + \frac{249.64}{24}} = \sqrt{76.27 + 10.40} = \sqrt{86.67} \approx 9.31$

step 4

Determine the z-score for a 90% confidence interval, which is approximately 1.645 for a two-tailed test

step 5

Calculate the margin of error (ME) using the formula $ME = z \times SE$

step 6

Substitute the values into the formula: $ME = 1.645 \times 9.31 \approx 15.34$

step 7

Calculate the confidence interval using the formula $CI = (\bar{x}_{\mathrm{R}} - \bar{x}_{\mathrm{L}}) \pm ME$

step 8

Substitute the values into the formula: $CI = (116.8 - 115.3) \pm 15.34 = 1.5 \pm 15.34$

step 9

Calculate the lower and upper bounds of the confidence interval: Lower bound $= 1.5 - 15.34 = -13.84$, Upper bound $= 1.5 + 15.34 = 16.84$

step 10

Round the values to two decimal places if necessary: CI $= (-13.84, 16.84)$

D

Key Concept

Confidence Interval for the Difference in Means

Explanation

The confidence interval for the difference between two means is calculated using the standard error of the difference, the appropriate z-score for the confidence level, and the sample means. The margin of error is then applied to the difference in sample means to find the interval.

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