a random sample of 10 minature tootsie rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were 3.087, 3.131,3.241,3.270,3.353,3.400,3.411,3.437,3.477 a.) construct a 90 percent confidence intervalfor the true meam weight. b.)what sample size would be necessary to estimate the true weight with an error of ± 0.03 grams with90 percent confidence?

a) To construct a 90% confidence interval for the true mean weight, we can use the formula:

Confidence Interval = Sample Mean ± (Critical Value * Standard Error)

First, let's calculate the sample mean:
Sample Mean = (3.087 + 3.131 + 3.241 + 3.270 + 3.353 + 3.400 + 3.411 + 3.437 + 3.477) / 10 = 32.039 / 10 = 3.2039 grams

Next, we need to calculate the standard deviation (SD) of the sample values:
1. Calculate the sample variance (S^2):
Sample Variance = (sum of (Observation - Sample Mean)^2) / (n - 1)
= [(3.087 - 3.2039)^2 + (3.131 - 3.2039)^2 + ... + (3.477 - 3.2039)^2] / (10 - 1)
= 0.3631537

2. Calculate the standard deviation (SD):
SD = sqrt(Sample Variance) = sqrt(0.3631537) ≈ 0.6026

We need to find the critical value (z-value) for a 90% confidence level. Since the sample size is small (n < 30), we use the t-distribution instead of the z-distribution. With 10 observations and a 90% confidence level, the degrees of freedom (df) is 10 - 1 = 9. From the t-distribution table or a statistical calculator, the critical value for a 90% confidence level with df = 9 is approximately 1.833.

Now, we can calculate the standard error:
Standard Error = SD / sqrt(n) = 0.6026 / sqrt(10) ≈ 0.1906

Finally, we can construct the 90% confidence interval:
Confidence Interval = Sample Mean ± (Critical Value * Standard Error)
= 3.2039 ± (1.833 * 0.1906) ≈ 3.2039 ± 0.3498

Therefore, the 90% confidence interval for the true mean weight is (2.8541 grams, 3.5537 grams).

b) To determine the sample size necessary to estimate the true weight with an error of ± 0.03 grams with 90% confidence, we can use the following formula:

Sample Size = (Critical Value * Standard Deviation / Margin of Error)^2

First, let's calculate the critical value corresponding to a 90% confidence level. Since we need to estimate the sample size, we will use the z-distribution as the sample size is not specified:

Using a z-table or calculator, the critical value for a 90% confidence level is approximately 1.645.

Next, we use the known standard deviation, which is 0.6026 grams, and the desired margin of error, which is 0.03 grams, in the formula:

Sample Size = (1.645 * 0.6026 / 0.03)^2 ≈ 109.5867

Therefore, a sample size of approximately 110 would be necessary to estimate the true weight with an error of ± 0.03 grams with 90% confidence.

To construct a confidence interval for the true mean weight of the miniature Tootsie Rolls, we can use the given sample data. Here's how you can find the answer step by step:

a) Calculating the 90 percent confidence interval for the true mean weight:

Step 1: Calculate the sample mean.
Add up all the weights and divide the sum by the sample size (10):
Sample Mean = (3.087 + 3.131 + 3.241 + 3.270 + 3.353 + 3.400 + 3.411 + 3.437 + 3.477) / 10 = 32.807 / 10 = 3.2807 grams.

Step 2: Calculate the standard deviation (SD) of the sample.
You can use either the sample standard deviation or the population standard deviation formula, depending on the context. In this case, we'll assume it's a sample, so we'll use the sample standard deviation formula:
a. Calculate the squared difference for each weight by subtracting the sample mean from each weight and squaring the result.
(3.087 - 3.2807)² = 0.03726161
(3.131 - 3.2807)² = 0.02247609
(3.241 - 3.2807)² = 0.00156571
(3.270 - 3.2807)² = 0.00011316
(3.353 - 3.2807)² = 0.00524121
(3.400 - 3.2807)² = 0.01427609
(3.411 - 3.2807)² = 0.01700249
(3.437 - 3.2807)² = 0.02435209
(3.477 - 3.2807)² = 0.03856661

b. Sum up all the squared differences.
Sum = 0.03726161 + 0.02247609 + 0.00156571 + 0.00011316 + 0.00524121 + 0.01427609 + 0.01700249 + 0.02435209 + 0.03856661 = 0.16085406

c. Divide the sum by (n-1) (where n is the sample size) to calculate the sample variance.
Sample Variance (s²) = Sum / (n - 1) = 0.16085406 / (10 - 1) = 0.16085406 / 9 = 0.01787267

d. Take the square root of the sample variance to get the sample standard deviation.
Sample Standard Deviation (s) = √(Sample Variance) = √(0.01787267) = 0.13371199

Step 3: Determine the critical value.
To establish a 90 percent confidence interval, we need to find the critical value associated with a confidence level of 90 percent and a sample size of 10. Consulting a t-distribution table (or using statistical software), the critical value for a 90 percent confidence level with 9 degrees of freedom is approximately 1.833.

Step 4: Calculate the margin of error (ME).
The margin of error is equal to the critical value multiplied by the sample standard deviation divided by the square root of the sample size.
ME = Critical Value * (Sample Standard Deviation / √Sample Size)
ME = 1.833 * (0.13371199 / √10) ≈ 0.0893

Step 5: Construct the confidence interval.
The confidence interval can be found by subtracting and adding the margin of error to the sample mean.
Confidence Interval = Sample Mean ± Margin of Error
Confidence Interval = 3.2807 ± 0.0893

Therefore, the 90 percent confidence interval for the true mean weight of the miniature Tootsie Rolls is approximately (3.2807 - 0.0893)g to (3.2807 + 0.0893)g, or 3.1914g to 3.3700g.

b) To estimate the true weight with an error of ± 0.03 grams and with 90 percent confidence, we need to find the required sample size.

The formula to calculate the sample size needed for a desired margin of error is:
Sample Size = (Critical Value * Standard Deviation / Desired Margin of Error)²

Rearranging the formula, we get:
Sample Size = (Critical Value * Standard Deviation / Desired Margin of Error)²

Substituting the given values:
Critical Value = 1.833 (from the t-distribution table for a 90 percent confidence level with 9 degrees of freedom)
Standard Deviation = 0.13371199 (calculated earlier)
Desired Margin of Error = 0.03 (given)

Sample Size = (1.833 * 0.13371199 / 0.03)²

By calculating this expression, you will find the required sample size for estimating the true weight with a desired error of ± 0.03 grams and 90 percent confidence.