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Determine whether the z-test or the t-test should be performed, or whether a statistician should be consulted.

A simple random sample of size 23 has mean x=14.3 and the standard deviation is s=2.45. The population is not approximately normal. Can you conclude that the population mean is less than 10?

The population standard deviation [is or isnt] known.
The sample size n [is or isnt] greater than 30.
The population [is or isnt] approximately normal.
The correct decision is to [perform the z-test? perform the t-test? or talk to a statistician?].

5 answers

To determine which statistical test to use, let's evaluate the provided information step by step:

  1. Population Standard Deviation: You mentioned that the population standard deviation is not known. This is important because the t-test is generally used when the population standard deviation is unknown.

  2. Sample Size: The sample size (\( n = 23 \)) is not greater than 30. Typically, when the sample size is small (less than 30), the t-test is preferred, especially if the population is not approximately normal.

  3. Normality of the Population: The population is not approximately normal. Since the sample size is small and the population distribution is not normal, this adds further emphasis on using the t-test.

Given these points, the correct decision is to:

  • The population standard deviation is not known.
  • The sample size \( n = 23 \) is not greater than 30.
  • The population is not approximately normal.

Therefore, the correct decision is to talk to a statistician since the conditions for performing a t-test are not satisfied due to the non-normality of the population and the small sample size. Using a t-test could lead to unreliable results in this scenario.

A simple random sample of size 13 has mean x=9.4 and standard deviation s=1.9. The population is approximately normally distributed. Can you conclude that the population mean differs from 11?
The population standard deviation [is or isnt?] known.
The sample size n [is or isnt] greater than 30.
The population [is or isnt?] approximately normal.
The correct decision is to [perform the z-test? perform the t-test? or talk to a statistician?].

To determine which statistical test to use, let's evaluate the provided information:

  1. Population Standard Deviation: It is indicated that the population standard deviation is not known. Therefore, since we don't have this information, we typically use the t-test when the population standard deviation is unknown.

  2. Sample Size: The sample size \( n = 13 \) is not greater than 30. When the sample size is small, the t-test is generally preferred, especially when the population is normally distributed.

  3. Normality of the Population: The population is stated to be approximately normal. This is crucial because the t-test is appropriate to use when the sample size is small and the population is approximately normally distributed.

Given these points, the correct decision is:

  • The population standard deviation is not known.
  • The sample size \( n = 13 \) is not greater than 30.
  • The population is approximately normal.

Therefore, the correct decision is to perform the t-test.

A simple random sample of size 28 has mean x=41.8. The population standard deviation is σ=3.8. The population is normally distributed. Can you conclude that the population mean is greater than 40?
The population standard deviation [is or isnt] known.
The sample size n [is or isnt] greater than 30.
The population [is or isnt] approximately normal.
The correct decision is to [perform the z-test? perform the t-test? or talk to a statistician?].

To determine which statistical test to use, let's evaluate the provided information:

  1. Population Standard Deviation: It is indicated that the population standard deviation is known (\( \sigma = 3.8 \)). This allows for the use of the z-test.

  2. Sample Size: The sample size \( n = 28 \) is not greater than 30. However, since the population standard deviation is known, we can still proceed with the z-test as long as the population is normally distributed.

  3. Normality of the Population: The population is stated to be normally distributed. This is important because it supports the use of the z-test, particularly when the population standard deviation is known.

Based on this information, we can conclude:

  • The population standard deviation is known.
  • The sample size \( n = 28 \) is not greater than 30.
  • The population is normally distributed.

Given these points, the correct decision is to perform the z-test.