Instructions: You must respond to at least 2 other students. Responses may include direct questions.

In your first peer posts, pick another confidence level, i.e. 90%, 99%, 97%, any other confidence level is fine.  Have fun and be creative with it and calculate another T-confidence interval and interpret your results.  Compare your results to that of the initial 95%, how much do they differ?  How useful can this type of information be when you go to buy a new car, or even a house?  

In your second peer posts, pick another confidence level, i.e. 90%, 99%, 97%, any other confidence level is fine.  Have fun and be creative with it and calculate another proportion interval and interpret your results.  Compare your results to that of the initial 95%, how much do they differ?  How useful can this type of information be when you go to buy a new car, or even a house?  

Make sure you include your data set in your initial post as well studen1  ruocco

Confidence intervals and sample size

Initial Confidence Intervals

From the provided data (excluding the supercar outlier), we have been given the mean, standard deviation, and sample size as follows;

Mean (µ) = 25014

Standard deviation (σ) = 88.47

Sample Size (n) = 10

We will first calculate the two T-confidence intervals at 95% from the given values. One interval will be a T-confidence interval for the sample mean, and the other will be a proportion confidence interval for the proportion of cars below the average price.

T-Confidence Interval at 95%

Calculating the T-confidence interval at 95% involves first calculating the T-critical value, then calculating the standard error (SE), calculating the Margin of Error (ME), and finally calculating the Confidence Interval.

Calculate the T-Critical Value

Confidence Level = 95%

Alpha (α) = 0.05

Degrees of Freedom (DF) = n – 1 = 10 – 1 = 9

T-Critical Value (T*) using Excel: =T.INV(0.975, 9)

Calculate the Standard Error (SE)

SE if calculated by taking Standard deviation divided by the square root of the sample size;

????????=3488.4710 divided by the square root of 10=1103.60

Calculate the Margin of Error (ME)

In calculating the ME, we consider the product of T-critical value (T) and the SE. Here, the T-critical value is 2.262, obtained from statistical tables o, and the Standard Error, calculated earlier, is 1103.60. The multiplication then gives us ME to be 2495.50 as shown;

????????=????*×????????=2.262×1103.60=2495.50

Calculate the Confidence Interval (CI)

Therefore, to calculate CI, we start with the mean, which represents the average value of our data set, in this case,014. The ME is determined by how much uncertainty there is in the estimate (Chang et al.), which, in this case, is calculated to be 495.50. We calculate the lower and upper bounds to find the range of true values.

The lower bound = ME minus mean= 518.50.

The upper bound = ME added to the mean= 509.50.

We are 95% confident that the actual mean price of the cars in the sample population lies between 518.50 and 509.50. This means that if we were to take many samples and calculate a confidence interval for each, approximately 95% of those intervals would contain the accurate mean price of the cars.: Confidence intervals and sample size

Proportion Confidence Interval at 95%

In calculating the potential confidence interval, we first calculate the p and q, then find the critical value, which is used to find the SE and ME.

Calculations

Calculate p and q

Number of cars below the average price: 4 (assuming p = 0.4)

????​=0.4

Therefore, ????=1−????=1−0.4=0.6

Calculate the Z-Critical Value

Confidence Level = 95%

Alpha (α) = 0.05

 For a 95% confidence level, we seek the Z* value corresponding to the cumulative probability of 0.975. We utilize the NORM.S.INV function in Excel, inputting the desired cumulative probability (0.975) to obtain the Z-critical value. 

Calculate the Standard Error (SE)

Calculating the SE involves first finding the multiplication of p and q values and then dividing the product by a number of intervals (Riley et al.). We then finally find the square root of the final value. 0.4 *0.6, then divide by 10 t to give 0.024. Its square root then gives us 0.1549

Calculate the Margin of Error (ME)

 The Margin Error is calculated by  taking the value of Z and multiplying it with SE value to give 0.3036

????????=1.96×0.1549=0.3036

Calculate the Confidence Interval

            The confidence interval is calculated by taking the p values, adding them to ME, and then negating the same value from ME.

????????=????±????????=0.4±0.3036C to find

???????????????????? ????????????????????=0.4−0.3036=0.0964

???????????????????? ????????????????????=0.4+0.3036=0.7036

Therefore  95% of cars are below the average price of between 9.64% and 70.36% hence imply that if we were to take many samples and calculate a confidence interval for each, approximately 95% of those intervals would have the proper proportion of cars below the average price.

Response 1: T-Confidence Interval at 90%

Calculate the T-Critical Value for 90%:

Confidence = 90%

 (α) = 0.10

 (DF) = 9

T-Critical Value (T*) using Excel: =T.INV(0.95, 9)

Calculate the new Margin of Error (ME):

????????=????*×????????=1.833×1103.60=2024.09

Calculate the new Confidence Interval:

????????=????±????????=25014±2024.09

???????????????????? ????????????????????=25014−2024.09=22989.91

????????????????????=25014+2024.09=27038.09

The result gives a 90% confident that exact mean price of the cars lies between $22,989.91 and $27,038.09. This narrower interval suggests we are less confident than the 95% confidence level. However, we still expect the true mean to fall within this range with high confidence.

 

student 2 

For the first interval you need to calculate a T-confidence interval for the sample population.

T confidence interval is the probability =CONFIDENCE.T(alpha, standard_dev, size 

My Mean was 176589

SD: 355498

95% T-CONFIDENCE

When I plug in the formula, =confindence.t (1,355498,13) I get a NUM# error. Once I tried it again, I got 165492.8783,518672.6965]  I am not sure what I did wrong the first time but im happy I at least got some type of number. 

Confidence Interval=176589.9091±(1.96×13​355498.348​)

PT2: 

Proportion=Total Sample SizeNumber of cars below average​

Proportion=Number of cars below average13Proportion=13Number of cars below average​

=0.191998,0.730002 

Proportion=0.461 total Sample Size=13

Shedrece