A consumer magazine has contacted a simple random sample of 33 owners of a certain model of automobile and asked each owner how many defects had to be corrected with in the first 2 months of ownership. The average number of defects was x with (with line over the x)=3.7, with a standard deviation of 1.8 defects.
a. Use the t distrubution to construct a 95% confidence interval for u = the average number of defects for this model.
b. Use the z distrubution to construct 95% confidence interval for u = the average number of defects for this model.
c. Given that the population standard deviation is not known, which of these two confidence intervals should be used as the interval estimate for u?
Use confidence interval formulas to construct the interval for µ.
CI95 = mean + or - 1.96 (sd/√n)
...where + or - 1.96 represents the 95% interval using a z-distribution, sd = standard deviation, and n = sample size.
Confidence interval using a t-distribution would be the same EXCEPT you will need to look at a t-table using degrees of freedom (which is n-1) for the + or - value.
You should be able to determine which confidence interval should be used.
I hope this will help get you started.
gvtu
a. To construct a 95% confidence interval for the population average number of defects (u), we will use the t-distribution.
Step 1: Determine the critical value of t.
Since our sample size is 33, we have 32 degrees of freedom (df = n - 1). Look up the critical value of t for a 95% confidence level and 32 degrees of freedom. Let's assume the critical value is t*.
Step 2: Calculate the standard error of the mean (SEM).
The standard error of the mean is calculated using the formula: SEM = standard deviation / square root of sample size. In this case, the standard deviation is 1.8 and the sample size is 33. So, SEM = 1.8 / sqrt(33).
Step 3: Calculate the margin of error (ME).
The margin of error is calculated by multiplying the SEM by the critical value: ME = t* * SEM.
Step 4: Calculate the confidence interval.
The confidence interval is calculated by subtracting the margin of error from the sample mean and adding it to the sample mean: Confidence Interval = sample mean - ME to sample mean + ME.
b. To construct a 95% confidence interval for the population average number of defects (u) using the z-distribution, we need to assume that the population standard deviation is known. However, in this case, the population standard deviation is not known. Therefore, using the z-distribution would not be appropriate.
c. Since the population standard deviation is not known, we should use the confidence interval constructed using the t-distribution (part a) as the interval estimate for u.