Question

1. What is a confidence-interval estimate of a parameter? Why is such an estimate superior to a point estimate?

2. Describe when you would use a t-score instead of a z-score when finding a confidence interval.

3. Suppose you want to increase the level of confidence without reducing the precision. What can you do?

Thanks in advance!

Answer 1

1)

A confidence interval is an interval which will capture the true mean of the population a certain percentage of the times it is computed. For example if we were to compute a 95% confidence interval of (10,15) we could say that if we computed this interval 100 times 95 out of that hundred the true population mean would have been captured in this interval. IIt is superior to a point estimate because with a point estimate we have no idea how close to the true population mean we are.

2)

Z- score :

Simply put, the z-score is a ‘standard score’. A z - score is the number of standard deviations from the mean of data point. But we can say broadly that z- score is a measure of how many standard deviations below or above the population mean a raw score is. it can be placed on a normal distribution curve.The basic z-score formula is-

z = (x – μ) / σ

Where: x= data point , μ= avareg score, σ= standard deviation.

We are well I can find several sources to understand and learn with examples and illustrations There are several web sites associated with this site. One of them is ZTable.

7 10.00 -3.6.0002 -3.5.0002 3.4 10003 -3.31.0005 -3.21.0007 -3.1 | .0010 -3.0] 0013 -29] .0019 -2.8.0026 -2.7.0035 -26].0047 -2.5.0062 -2.4.0082 -2.3 | 20107 -2.21.0139 -21 20179 -20].0228 -1.9 0287 0.01 1.0002 1.0002 1.0003 1.0005 1.0007 1.0009 1.0013 1.0018 1.0025 1.0034 1.0045 1.0060 1.0080 .01 1. 66 174 1.0222 10281 0.02 .0001 1.0002 1.0003 1.0005 .0006 .0009 1.00 13 1.00 18 1.0024 1.0033 0044 20059 10078 20102 .0132 20170 1.0217 0274 0.03 10001 10002 10003 1.004 .0006 .009 .0012 10017 1.0023 .0032 1.0043 .0057 .0075 .0099 20129 .0166 1.0212 1.0268 0.04 1,0001 1,0002 1,0003 1.0004 1,0006 1.0008 1.0012 1.00 16 1.0023 0031 1.0041 .0055 1.0073 1.0096 1.0125 20162 ,0207 0262 0.05 1.0001 .0002 .0003 .0004 .0006 .0008 0011 40016 10022 .0030 1.0040 .0054 .0071 .0094 20122 20158 10202 1.0256

T-score:

A t-score is one form of a standardized test statistic. The t-score formula enables to take an individual score and transform it into a standardized form one which helps I to compare scores.I will want to use the t-score formula when I don’t know the population standard deviation. The t- score formula is -

t= I-up S n

Where ; x̄ = sample mean, μ0 = population mean, s = sample standard deviation, n = sample size. For know details visit the site ZTable.org.

Difference between z-score and t-score:

A z-score and a t-score are both used in hypothesis testing. Few topics in elementary statistics cause more confusion to learners than deciding when to use the z-score or t-score. Usually in elementary stats and AP stats, we’ll use a z-score in testing more often than a t-score.Like the z-scores, the t-scores are also a conversion of individual scores into a standard form. However, t-scores are used when we don’t know the population standard deviation; we make an estimate by using our sample.

3)

Answer: We can increase the level of confidence without reducing the precision by increasing the sample size n. Here the precision of the estimate is fixed. i.e. length of the confidence interval 2zą is fixed. Now we want to increase the confidence level (1-a)i.e. equivalently we want to increase the value of <a since the value of 2ą increases as the confidence level (1-a)increases. But we 2 2 have to keep the value of 2za constant. Therefore we have to increase the value of nin the Vn denominator such that the quantity 2za remains constant. Hence n this way we can ✓n increase the confidence level without affecting the precision of the estimate.