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Determine whether we can safely use a t critical value to calculate a confidence interval for the population mean in each of the following settings:

a) A safety specialist was concerned about cars' speeds on a certain busy stretch of road. They randomly selected 21 cars and measured their speeds at the same point on the road. Each car was from a different time period and day, so the specialist is willing to consider the sample as representative. The speeds in the sample were strongly skewed to the left with a sample mean of 96.5 km/hr. [ Select ] ["no", "yes"]

b) Layla is working on a PhD in computer science and wonders what the average page length is for dissertations in her field. She takes a random sample of 80 computer science dissertations and notices that the distribution of their page lengths is skewed to the right with a sample mean of 125.4 pages. [ Select ] ["yes", "no"]

c) Rex wanted to estimate the mean finishing time for the approximately 50,000 finishers at the New York City Marathon. He took a random sample of 10 finishers' times, which were approximately normal with a sample mean of 271.5 minutes. He's considering using his data to make a confidence interval for the mean finishing time. [ Select ] ["no", "yes"]

d) Nolan was studying birth weights of infants in Somalia. He took an SRS (simple random sample) of 100 births and calculated a sample mean birth weight of 3.2 kg. The sample data was slightly skewed to the right. He is considering using his data to construct a confidence interval for the overall percent of babies in Somalia that are underweight. [ Select ] ["no", "yes"]

Answer :

to calculate a confidence interval for the population mean in each of the following setting conditions are as follows a) No b) Yes c) No and d) No

In scenario (a), the sample speeds are strongly skewed to the left, indicating a departure from normality. As the sample size is relatively small (n = 21), violating the assumption of normality raises concerns about the validity of using a t-distribution critical value. Therefore, a t-distribution critical value should not be used in this case.

In scenario (b), although the distribution of page lengths is skewed to the right, the sample size is relatively large (n = 80). With a larger sample size, the central limit theorem suggests that the sampling distribution of the sample mean becomes more approximately normal. Thus, it is safe to use a t-distribution critical value to calculate a confidence interval for the population mean.

In scenario (c), the sample size is small (n = 10) and there is no information about the shape of the population distribution. Hence, assuming normality may not be appropriate, and a t-distribution critical value should not be used.

In scenario (d), although the sample data is slightly skewed right, there is no mention of the sample size. Without information about the sample size, it is not possible to determine whether a t-distribution critical value can be used. However, constructing a confidence interval for the overall percent of babies in Somalia that are underweight would require a different approach, such as using a proportion or binomial distribution.

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