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NO LINKS!! Please help me with this problem 1b​

NO LINKS!! Please help me with this problem 1b​-example-1

2 Answers

3 votes

Answer:


\textsf{b)} \quad 15.8 \pm 0.17

Step-by-step explanation:

Given values:

  • Sample size n = 200
  • Population standard deviation σ = 1.2
  • Sample mean
    \overline{x} = 15.8


\boxed{\begin{minipage}{7 cm}\underline{Standard error of the mean}\\\\$SE=(\sigma)/(√(n))$\\\\where:\\\phantom{ww}$\bullet$ $\sigma$ is the population standard deviation. \\ \phantom{ww}$\bullet$ $n$ is the sample size.\\\end{minipage}}

Substitute the given values into the standard error formula:


\implies SE=(1.2)/(√(200))=(3√(2))/(50)

The critical value for a 95% confidence level using normal distribution is:


z=1.9600\;\;\sf (4\;d.p.)


\boxed{\begin{minipage}{6.7 cm}\underline{Margin of error}\\\\$ME=z \cdot SE$\\\\where:\\\phantom{ww}$\bullet$ $z$ is the critical value. \\ \phantom{ww}$\bullet$ $SE$ is the standard error of the mean.\\\end{minipage}}

Substitute the given values into the margin of error formula:


\implies ME=1.9600 \cdot (3√(2))/(50)


\implies ME=0.17\;\; \sf (2\;d.p.)

Therefore, an estimate for the size of the population mean, including margin or error is:


\implies \mu=\overline{x}\pm ME


\implies \mu=15.8 \pm 0.17

User Hrant Nurijanyan
by
7.5k points
0 votes

Answer: Choice B)
\boldsymbol{15.8 \ \pm \ 0.17}

======================================================

Step-by-step explanation:

sigma = 1.2 = population standard deviation

n = 200 = sample size

Your teacher doesn't mention the confidence level, so I'll assume we go for the default 95%.

At 95% confidence, the z critical value is roughly z = 1.96

The margin of error (E) is calculated like so:

E = z*sigma/sqrt(n)

E = 1.96*1.2/sqrt(200)

E = 0.1663 approximately

That rounds to 0.17

Therefore the estimate for the population mean in the format
\overline{\text{x}} \ \pm \ E is
\boldsymbol{15.8 \ \pm \ 0.17} which is the final answer.

------------

This is the shorthand way of saying the mean mu is between
\overline{\text{x}} \ - \ E = 15.8 - 0.17 = 15.63 and
\overline{\text{x}} \ + \ E = 15.8 +0.17 = 15.97

In other words,


15.63 < \mu < 15.97 at 95% confidence.

User Sameer Kazi
by
8.3k points

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