Olivia works at a company that creates mobile phones. She wanted to estimate the mean amount of time their new phone's battery lasts with regular use after a full charge. She took a random sample of 6 of these phones and randomly assigned each of them to a volunteer. She instructed them to fully charge the phones and use them as they regularly would until the battery died (without recharging the phone). Here are the data they reported:
\begin{tabular}{rrrrrrr}
Phone & 1 & 2 & 3 & 4 & 5 & 6 \
\hline Battery life (hours) & 8.0 & 6.0 & 10.5 & 9.0 & 8.5 & 12 \
Mean & $\bar{x}=9$ hours & & & & &
\end{tabular}

Standard deviation $\quad s_{x}=2.07$ hours

Assume that all conditions for inference are met.
Which of the following is a $90 \%$ confidence interval for the mean battery life (in hours)?

Choose 1 answer:
(A) $9 \pm 1.4$
(B) $9 \pm 1.7$
(C) $9 \pm 2.0$
(D) $9 \pm 2.07$

Question

Olivia works at a company that creates mobile phones. She wanted to estimate the mean amount of time their new phone's battery lasts with regular use after a full charge. She took a random sample of 6 of these phones and randomly assigned each of them to a volunteer. She instructed them to fully charge the phones and use them as they regularly would until the battery died (without recharging the phone). Here are the data they reported:
\begin{tabular}{rrrrrrr} 
Phone & 1 & 2 & 3 & 4 & 5 & 6 \
\hline Battery life (hours) & 8.0 & 6.0 & 10.5 & 9.0 & 8.5 & 12 \
Mean & $\bar{x}=9$ hours & & & & &
\end{tabular}

Standard deviation $\quad s_{x}=2.07$ hours

Assume that all conditions for inference are met.
Which of the following is a $90 \%$ confidence interval for the mean battery life (in hours)?

Choose 1 answer:
(A) $9 \pm 1.4$
(B) $9 \pm 1.7$
(C) $9 \pm 2.0$
(D) $9 \pm 2.07$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To calculate the 90% confidence interval for the mean battery life, we use the formula for the T-interval: $x̄ ± (t_{(1 - c)/2} \cdot s)/\sqrt{n}$, where $x̄$ is the sample mean, $s$ is the sample standard deviation, $n$ is the sample size, and $c$ is the confidence level. ‖ ‖ step 2 ⋮ Given that the sample mean $x̄ = 9$ hours, the sample standard deviation $s = 2.07$ hours, the sample size $n = 6$, and the confidence level $c = 90\%$, we can plug these values into the formula. ‖ ‖ step 3 ⋮ The critical value $t_{(1 - c)/2}$ for a 90% confidence interval with 5 degrees of freedom (since $n - 1 = 6 - 1 = 5$) can be found in a t-distribution table or using a calculator. ‖ ‖ step 4 ⋮ The calculation yields a confidence interval of $9 ± 2.07$. This matches option (D) from the multiple-choice answers provided. ‖ ***D*** ∻Key Concept∻ ⚹T-interval for a population mean⚹ ∻Explanation∻ ⚹The T-interval formula is used to estimate the population mean from a sample mean, standard deviation, and size, with a given confidence level.⚹◊What is the formula to calculate the margin of error for a confidence interval in statistics?⍭ Generate me a similar question◊

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