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Answer :
To solve this question, we need to understand what a "two-tailed test at the 5% level" means in the context of statistics.
A two-tailed test is used when you are interested in determining whether a sample is significantly different from the population mean in either direction (higher or lower).
The significance level of 5% means that there is a total probability of 0.05 allocated to the two tails of the normal distribution. This is split equally between the two tails, with 0.025 (2.5%) in each tail.
In a standard normal distribution, the critical Z scores represent the values beyond which we would reject the null hypothesis. For a two-tailed test at the 5% level:
1. You look up the critical Z score that corresponds to 0.025 in the lower tail. This will be the negative critical value.
2. Similarly, for the upper tail, you look at the 0.975 (because 1 - 0.025 = 0.975) cumulative probability in the standard normal distribution, which gives you the positive critical value.
From statistical Z tables or calculators:
- The Z score that corresponds to 0.025 in the lower tail is approximately -1.96.
- The Z score that corresponds to 0.975 in the upper tail is approximately 1.96.
Thus, in a two-tailed test at the 5% level, the critical boundary Z scores are +1.96 and -1.96.
A two-tailed test is used when you are interested in determining whether a sample is significantly different from the population mean in either direction (higher or lower).
The significance level of 5% means that there is a total probability of 0.05 allocated to the two tails of the normal distribution. This is split equally between the two tails, with 0.025 (2.5%) in each tail.
In a standard normal distribution, the critical Z scores represent the values beyond which we would reject the null hypothesis. For a two-tailed test at the 5% level:
1. You look up the critical Z score that corresponds to 0.025 in the lower tail. This will be the negative critical value.
2. Similarly, for the upper tail, you look at the 0.975 (because 1 - 0.025 = 0.975) cumulative probability in the standard normal distribution, which gives you the positive critical value.
From statistical Z tables or calculators:
- The Z score that corresponds to 0.025 in the lower tail is approximately -1.96.
- The Z score that corresponds to 0.975 in the upper tail is approximately 1.96.
Thus, in a two-tailed test at the 5% level, the critical boundary Z scores are +1.96 and -1.96.
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