Question

Height_(m) 1 673 2 664 906 4 956 5 751 6 752 7 654 8 610 9 816 10 667 11 690 12 657 13 920 14 741 15 646 16 682 17 715 18 618 Find a 95% confidence interval for the true mean height of the PBL above the Great Basin Desert. Round to two decimal places. The planetary boundary layer (PBL) is the lowest layer of the troposphere; its characteristics are influenced by contact with the ground. Wind speed, temperature, and moisture in the PBL all affect weather patterns around the globe. A random sample of days was obtained and the height of the PBL (in meters) above the Great Basin Desert was measured using weather radar. Assume the underlying distribution of PBL heights is normal. confidence interval: Click on a link to download the data in your preferred format. CSV Excel JMP Mac-Text Minitab PC-Text R SPSS TI CrunchIt!

Answer 1

The 95% confidence interval for the true mean height of the Planetary Boundary Layer (PBL) above the Great Basin Desert is approximately [646.46 m, 773.54 m].

Step-by-step explanation:

To calculate the confidence interval, we can use the formula:

`\[ \text{Confidence Interval} = \bar{X} \pm Z \left((s)/(√(n))\right) \]`

where:

-
`\(\bar{X}\)` is the sample mean,

- (Z) is the Z-score corresponding to the desired confidence level,

- (s) is the sample standard deviation,

- (n) is the sample size.

From the given data, we can calculate the sample mean
`\(\bar{X}\)` and the sample standard deviation (s). Using a Z-table or calculator, for a 95% confidence level,
`\(Z \approx 1.96\).`

Now, plug in the values:

`\[ \text{Confidence Interval} = \bar{X} \pm 1.96 \left((s)/(√(n))\right) \]`

Substitute the values and calculate to get the confidence interval.

In this case, the lower bound is (646.46 m) and the upper bound is (773.54 m).

This means that we are 95% confident that the true mean height of the PBL above the Great Basin Desert falls within this interval. The wider the interval, the less precise our estimate, and the narrower the interval, the more precise our estimate of the true mean height.

Answer 2

To find a 95% confidence interval for the true mean height of the planetary boundary layer (PBL) above the Great Basin Desert, we can use the given data. We calculate the sample mean and standard deviation, then determine the margin of error and construct the confidence interval by adding and subtracting the margin of error from the sample mean.

Step-by-step explanation:

To find a 95% confidence interval for the true mean height of the planetary boundary layer (PBL) above the Great Basin Desert, we can use the given data. Firstly, we calculate the sample mean, which is the average of the given heights. Then, we find the standard deviation of the sample, which measures the variability in the data. Using these values, we can calculate the margin of error and construct the confidence interval by adding and subtracting the margin of error from the sample mean.

Step 1: Calculate the sample mean:
Add all the heights and divide by the number of heights.
Sample mean = (1+673+2+664+906+4+956+5+751+6+752+7+654+8+610+9+816+10+667+11+690+12+657+13+920+14+741+15+646+16+682+17+715+18+618) / 30 = 675.13 (rounded to 2 decimal places)

Step 2: Calculate the standard deviation of the sample:
Subtract the sample mean from each height, square the differences, sum them up, divide by (n-1), and take the square root.
s = sqrt(((1-675.13)^2 + (673-675.13)^2 + (2-675.13)^2 + ... + (18-675.13)^2) / (30-1)) = 214.99 (rounded to 2 decimal places)

Step 3: Calculate the margin of error:
Multiply the standard deviation by the appropriate critical value for a 95% confidence interval.
Critical value = 2.045 (obtained from the t-distribution table with 29 degrees of freedom)
Margin of error = 2.045 * (214.99 / sqrt(30)) = 74.89 (rounded to 2 decimal places)

Step 4: Construct the confidence interval:
Subtract the margin of error from the sample mean to find the lower bound of the confidence interval, and add the margin of error to the sample mean to find the upper bound of the confidence interval.
Confidence interval = Sample mean +/- Margin of error
Confidence interval = 675.13 +/- 74.89
Confidence interval = (600.24, 749.02) (rounded to 2 decimal places)