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The $95\%$confidence interval for the mean: $(430,470)$

Sample size (n)$=30$

We know, the formula of

Sample mean: $\stackrel{炉}{x}=\frac{\text{Lower limit}+U_{\text{pper limit}}}{2}$

The margin of error: $E=\frac{U_{pper}\text{limit-Lower limit}}{2}$

Standard error: $SE=\frac{s}{\sqrt{n}}$

Using the confidence interval, the Sample mean is,

$\stackrel{炉}{x}=\frac{430+470}{2}=450$

And the margin of error is,

$E=\frac{470-430}{2}=20$

First, find the critical-$t$-value:

$t_{伪/2}=2.042..\text{Using excel formula,}=TINV(0.05,29)$

The margin of error formula can be used to calculate the standard deviation.

$$\begin{gathered} E=t_{\frac{伪}{2}}脳\frac{s}{\sqrt{n}} \\ 鈬�20=2.042脳\frac{s}{\sqrt{30}} \\ 鈬�s=\frac{20}{0.3229} \\ 鈬�s=61.9447 \end{gathered}$$

standard deviation $=61.9447$

So, the standard error is$SE=\frac{61.9447}{\sqrt{30}}=11.3095$

Comment from a reporter: $95\%$of this company's AA batteries last between $430$and $470$minutes.

We can't be certain about the confidence interval because the battery will either last between $430$and $470$minutes or it won't. The correct interpretation is that we are $95\%$certain that the average battery life is between $430$and$470$minutes.

Claim: There are a $95\%$chance that the average life will be between $430$and $470$minutes.

We can't be certain about the confidence interval because the battery will either last between $430$and $470$minutes or it won't. If the sample means were very unlikely to occur by chance for a given population mean, the probability of a different sample means being in the same confidence interval is less than $95\%$ As a result, I disagree with the claim

We are $95\%$certain that the true population means is between $430$and$470$ minutes.