Given below are two statements:

Statement I: In two tail test, 2.5% of the area of the normal curve falls on the right of 1.96 standard deviations above the mean and 2.5% to the left of 1.96 standard deviations below the mean.

Statement II: In one tail test, for testing the difference between large sample means, a z score of 1.64 is taken as Yardstick at the 5% level of significance.

In light of the above statements, choose the most appropriate answer from the options given below: 

Explanation:

Statement 1: In two tail test, 2.5% of the area of the normal curve falls on the right of 1.96 standard deviations above the mean and 2.5% to the left of 1.96 standard deviations below the mean.

• A two-tailed test, in statistics, is a method in which the critical area of a distribution is two-sided and tests whether a sample is greater than or less than a certain range of values.
• It is used in null-hypothesis testing and testing for statistical significance. 
• 
• In the above figure,  Z of -2.5 represents a value 2.5 standard deviations below the mean with SD 1.96
• And Z of 2.5 represents a value of 2.5 standard deviations above the mean with SD 1.96.

Thus, statement 1 is true

Statement 2:  In one tail test, for testing the difference between large sample means, a z score of 1.64 is taken as Yardstick at the 5% level of significance.

• When a hypothesis test is set up to show that the sample mean would be higher or lower than the population means, this is referred to as a one-tailed test.
• The one-tailed test gets its name from testing the area under one of the tails (sides) of a normal distribution.
• The most common significance levels (p-values) used in a one-tailed test are 1, 5, and 10 percent.
• The critical z‐value for a probability of 0.05 in the upper tail is 1.64 or 1.65.

Thus, both statements 1 and 2 are correct.

Additional Information: Z score:

• A Z-score numerically describes a value's relationship to the mean of a group of values and is measured in terms of the number of standard deviations from the mean.

  • If a Z-score is 0, it indicates that the data point's score is identical to the mean score whereas Z-scores of 1.0 and -1.0 would indicate values one standard deviation above or below the mean.
  • In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above and below the mean