High School

You intend to estimate a population mean \( \mu \) from the following sample:

54.8, 50.7, 55.1, 44.1, 43.2, 56.8, 50.8, 39.1, 47.2, 57, 59.7, 39.1, 42.9, 49.5, 49.2, 49.3, 61.4, 62.5, 52.9, 38.7, 59.1, 46.3, 61.7, 49.1, 47.8, 57.7, 45, 44.4, 43.9, 38.7, 74.7, 53.8, 42.5, 53.1, 46.6, 50.2, 47.7, 43.9, 55.5, 58.2, 58.8, 49.8, 65.7, 43.2, 49.4, 41.4, 25.6, 2.

Find the 99.9% confidence interval. Enter your answer as a tri-linear inequality accurate to two decimal places (because the sample data are reported accurate to one decimal place).

\(< \mu <\)

The answer should be obtained without any preliminary rounding. However, the critical value may be rounded to 3 decimal places.

Answer :

To calculate a 99.9% confidence interval for the population mean [tex]\mu[/tex], we need to follow a series of steps involving statistical concepts. Here’s how we can do it:


  1. Find the Sample Mean ([tex]\bar{x}[/tex]):
    First, we calculate the average of the given sample data.

    [tex]\bar{x} = \frac{\sum{x_i}}{n}[/tex]
    where [tex]x_i[/tex] is each value in the sample and [tex]n[/tex] is the number of sample values.

    Plugging in the numbers:
    [tex]\bar{x} = \frac{54.8 + 50.7 + 55.1 + \ldots + 25.6}{49}[/tex]

    After calculating the sum and division:
    [tex]\bar{x} \approx 49.89[/tex]


  2. Calculate the Sample Standard Deviation ([tex]s[/tex]):
    The sample standard deviation is calculated using:

    [tex]s = \sqrt{\frac{\sum{(x_i - \bar{x})^2}}{n-1}}[/tex]

    After computing the individual squared differences and the sum, you can find [tex]s \approx 9.26[/tex].


  3. Determine the Critical Value ([tex]Z[/tex]):
    For a 99.9% confidence interval with a normal distribution, we use a Z-distribution. The critical value [tex]Z[/tex] for 99.9% confidence level is approximately 3.291. This value is typically found in Z-tables.


  4. Calculate the Margin of Error (ME):
    The margin of error can be found using:

    [tex]ME = Z \times \frac{s}{\sqrt{n}}[/tex]

    [tex]ME \approx 3.291 \times \frac{9.26}{\sqrt{49}}[/tex]

    [tex]ME \approx 3.291 \times 1.32 \approx 4.35[/tex]


  5. Calculate the Confidence Interval:
    Finally, use [tex]ME[/tex] to find the confidence interval for [tex]\mu[/tex]:

    [tex](\bar{x} - ME) < \mu < (\bar{x} + ME)[/tex]

    [tex]49.89 - 4.35 < \mu < 49.89 + 4.35[/tex]

    [tex]45.54 < \mu < 54.24[/tex]



Thus, the 99.9% confidence interval for the population mean [tex]\mu[/tex] is approximately [tex]45.54 < \mu < 54.24[/tex]. This interval gives us a range that we can be 99.9% confident contains the true population mean.

Other Questions