**Old Faithful Eruption Times Analysis**Old Faithful, the most famous geyser in the world, located in Yellowstone National Park, has a mean time between eruptions of 85 minutes. The interval of time between eruptions is normally distributed with a standard deviation of 21.25 minutes. This exercise involves answering questions about the probability of certain time intervals based on this data.### Questions:**(a)** What is the probability that a randomly selected time interval is longer than 96 minutes?- Probability (rounded to four decimal places): [____]**(b)** What is the probability that a random sample of 10 time intervals between eruptions has a mean longer than 96 minutes?- Probability (rounded to four decimal places): [____]**(c)** What is the probability that a random sample of 24 time intervals between eruptions has a mean longer than 96 minutes?- Probability (rounded to four decimal places): [____]**(d)** What effect does increasing the sample size have on the probability? Provide an explanation for this result.- Fill in the blanks: - If the sample size increases, the probability [increases/decreases] because the variability in the [sample mean/population mean] [increases/decreases].**(e)** What might you conclude if a random sample of 24 time intervals between eruptions has a mean longer than 96 minutes? Select all that apply:- [ ] A. The population mean must be less than 85 minutes, since the probability is so low.- [ ] B. The population mean is 85 minutes, and this is just a rare sampling.- [ ] C. The population mean is 85 minutes, and this is an example of a typical sampling result.- [ ] D. The population mean must be more than 85 minutes, since the probability is so low.- [ ] E. The population mean may be less than 85 minutes.- [ ] F. The population mean may be greater than 85 minutes.- [ ] G. The population mean cannot be 85 minutes, since the probability is so low.**(f)** On a certain day, suppose there are 28 time intervals for Old Faithful. Treating these 28 eruptions as a random sample, there is a 0.20 likelihood that the mean length of time between eruptions will exceed what value?- The likelihood the mean length of time

As per our guidelines I can solve only first three subparts. Kindly post the remaining subparts again and mention them.Given that, population mean is u= 85 standard deviation is sd= 21.25 we know that, Z score for any sample is given by, z-score= (X-u)/sd and z score for sample mean is given by, z-score= (X-u)/sd/√n(a). P(X>96)= P(z>(96-85)/21.25) = P(z>0.47) = 1-P(z<0.47) From z score table, = 1-0.3192 = 0.6808 (b). Sample size is n= 10 Then, P(X bar> 96)= P(z>(96-85)/21.25/√10) = P(z>1.49) =1-P(z<1.49) From z score table, = 1-0.0681 = 0.9319 (C). Sample size is n= 24 Then, P(X bar> 96)= P(z>(96-85)/21.25/√24) = P(z>2.31) =1-P(z<2.31) From z score table, = 1-0.1044 = 0.9896

Math

Statistics

The most famous geyser in the world, Old Faithful in Yellowstone National Park, has a mean time between eruptions of 85 minutes. If the interval of time between the eruptions is normally distributed with standard deviation 21.25 minutes, complete parts (a) through (f). The probability that a randomly selected time interval is longer than 96 minutes is approximately. (Round to four decimal places as needed.) (b) What is the probability that a random sample of 10 time intervals between eruptions has a mean longer than 96 minutes? The probability that the mean of a random sample of 10 time intervals s more than 96 minutes is approximately (Round to four decimal places as needed.) (c) What is the probability that a random sample of 24 time intervals between eruptions has a mean longer than 96 minutes? The probability that the mean of a random sample of 24 time intervals is more than 96 minutes is approximately (Round to four decimal places as needed.) (d) What effect does increasing the sample size have on the probability? Provide an explanation for this result. Fill in the blanks below. If the sample size increases, the probability because the variability in the (e) What might you conclude if a random sample of 24 time intervals between eruptions has a mean longer than 96 minutes? Select all that apply. A. The population mean must be less than 85 minutes, since the probability is so low. B. The population mean is 85 minutes, and this is just a rare sampling. C. The population mean is 85 minutes, and this is an example of a typical sampling result. D. The population mean must be more than 85 minutes, since the probability is so low. E. The population mean may be less than 85 minutes. F. The population mean may be greater than 85 minutes. G. The population mean cannot be 85 minutes, since the probability is so low. mean The likelihood the mean length of time between eruptions exceeds minutes is 0.20. (Round to one decimal place as needed.) ▼ (f) On a certain day, suppose there are 28 time intervals for Old Faithful. Treating these 28 eruptions as a random sample, there is a 0.20 likelihood that the mean length of time between eruptions will exceed what value?