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Answer :
Final answer:
To construct a 90% confidence interval with a sample size of 37, one would need to find the t-score from a t-distribution table or calculator, which will be greater than 1.645. The provided options are incorrect, but 1.69 is close to the typical value used.
Explanation:
Finding the Critical t Value for a 90% Confidence Interval
When constructing a 90% confidence interval for a normally distributed population, the critical value needed is based on the Student's t-distribution, since the population standard deviation is unknown. Given a sample size of 37, the degrees of freedom (df) is 36, which is calculated as df = n - 1, where n is the sample size.
To find the critical value tα/2, also known as the t-score, we must look at a t-distribution table or use a calculator function such as invT. For a 90% confidence interval, a total of 10 percent is split into both tails of the distribution, being 5 percent in each tail.
Without access to the exact values of a t-distribution table or a calculator at hand, we cannot specify the exact value in this context. However, it is known that the value will be greater than the corresponding z-score for a normal distribution, which is 1.645 for a 90% confidence level.
Since the provided options (a, b, c, d) all mention the same value of 1.69 (which is an error in the question as they should be different), there must be a typo, and none of these values can be selected confidently as the correct one. In any case, 1.69 is very close to the correct value for a 90% confidence level and may be considered an approximation.
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