The standard deviation for a population is σ = 15.3. A sample of 36 observations selected from this population gave a mean equal to 74.8.

a) Make a 90% confidence interval for μ
b) Construct a 95% confidence interval for μ.
c) Determine a 99% confidence interval for μ.
d) Does the width of the confidence intervals constructed in parts a through c increase as the confidence level increases? Explain your answer

In a recent study, 25 males used a new weight-loss supplement, and all but 6 of them experienced weight loss after two weeks. In the same study, 100 females used the same supplement, and all but 31 of them experienced weight loss after two weeks.

Fill in the blanks of the statement below to make the statement the most reasonable possible.

% = mean ± ? SEm

SEm = SD/√n

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability ([1-%]/2) to find the Z score. Insert that into ? position.

Example: 95% = 74.8 ± 1.96 SEm

After you do a through c, you should be able to explain.

a) Well, constructing a 90% confidence interval for μ is like trying to find a good hiding spot for your secret recipe. You want it to be secure, but not too hidden that you forget where you put it. So, using the formula for a confidence interval, we can calculate it as:

Lower Bound = 74.8 - (1.645 * (15.3 / √36))
Upper Bound = 74.8 + (1.645 * (15.3 / √36))
Go ahead and crunch the numbers, Sherlock!

b) Now, let's crank it up a notch and construct a 95% confidence interval for μ. This is like wearing your favorite outfit and going to a party - you're confident and ready to show off! Use the same formula as before, but with a different z-value:
Lower Bound = 74.8 - (1.96 * (15.3 / √36))
Upper Bound = 74.8 + (1.96 * (15.3 / √36))
Get your dancing shoes on, because we're going to have a wider range of possibilities this time!

c) Alright, time to go all out and construct a 99% confidence interval for μ. This is like playing hide-and-seek with your friends in a massive mansion - you want to make sure you cover all the hiding spots before they find you! Use the same trusty formula, but with a different z-value:
Lower Bound = 74.8 - (2.576 * (15.3 / √36))
Upper Bound = 74.8 + (2.576 * (15.3 / √36))
Keep your eyes peeled, because this interval is going to be wider than an ostrich with its head in the sand!

d) Ah, the width of the confidence intervals! Think of it this way - as the confidence level increases, we become more and more certain about the range of possible values for μ. That certainty comes at a cost - the intervals will widen as we become more confident. So, yes, the width of the intervals will increase as the confidence level goes up. It's like opening your arms wider to receive more hugs - the more hugs you want, the more space you need!

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

Confidence Interval = (sample mean) ± (critical value) * (standard deviation / √sample size)

a) For a 90% confidence interval, the critical value can be found using a normal distribution table or a statistical calculator. Using either method, we find that the critical value for a 90% confidence level is approximately 1.645.
So, the 90% confidence interval is:
(74.8) ± (1.645) * (15.3 / √36) = 74.8 ± 3.192

Therefore, the 90% confidence interval for μ is (71.608, 78.992).

b) For a 95% confidence interval, the critical value can be found using a normal distribution table or a statistical calculator. Using either method, we find that the critical value for a 95% confidence level is approximately 1.96.
So, the 95% confidence interval is:
(74.8) ± (1.96) * (15.3 / √36) = 74.8 ± 3.403

Therefore, the 95% confidence interval for μ is (71.397, 78.203).

c) For a 99% confidence interval, the critical value can be found using a normal distribution table or a statistical calculator. Using either method, we find that the critical value for a 99% confidence level is approximately 2.576.
So, the 99% confidence interval is:
(74.8) ± (2.576) * (15.3 / √36) = 74.8 ± 4.139

Therefore, the 99% confidence interval for μ is (70.661, 78.939).

d) Yes, the width of the confidence intervals constructed in parts a through c increases as the confidence level increases. This is because a higher confidence level requires including a larger range of values around the sample mean to capture a larger proportion of the population. Consequently, a larger range results in a wider confidence interval.

To calculate confidence intervals, you can use the formula:

Confidence Interval = Sample Mean ± (Z * (σ / √n))

Where:
- Sample Mean is the mean of the sample observations
- Z is the Z-score that corresponds to the desired level of confidence
- σ is the standard deviation (known in this case)
- n is the sample size

a) To make a 90% confidence interval for μ:
- The level of confidence is 90%, so the corresponding Z-score is 1.645 (you can look up the Z-score from a Z-table).
- Substituting the values: Confidence Interval = 74.8 ± (1.645 * (15.3 / √36))
- Calculate the interval: Confidence Interval = 74.8 ± 4.03
- The 90% confidence interval for μ is (70.77, 78.83).

b) To construct a 95% confidence interval for μ:
- The level of confidence is 95%, so the corresponding Z-score is 1.96.
- Substituting the values: Confidence Interval = 74.8 ± (1.96 * (15.3 / √36))
- Calculate the interval: Confidence Interval = 74.8 ± 4.52
- The 95% confidence interval for μ is (70.28, 79.32).

c) To determine a 99% confidence interval for μ:
- The level of confidence is 99%, so the corresponding Z-score is 2.576.
- Substituting the values: Confidence Interval = 74.8 ± (2.576 * (15.3 / √36))
- Calculate the interval: Confidence Interval = 74.8 ± 6.15
- The 99% confidence interval for μ is (68.65, 80.95).

d) The width of the confidence intervals does increase as the confidence level increases. This is because as the level of confidence increases, the Z-score becomes larger, resulting in a larger margin of error. The larger margin of error leads to wider intervals, as it allows for a greater range of possible population means.