EduNinjaThe lengths, X cm, of a sample of 100 insects of a certain type were summarised as follows.
Calculate unbiased estimates for the population mean and variance of X.
State a necessary condition for the estimates found in part (a) to be reliable.
The time taken in minutes for a certain daily train journey has a normal distribution with standard deviation 5.8. For a random sample of 20 days the journey times were noted and the mean journey time was found to be 81.5 minutes.
Calculate a 98\% confidence interval for the population mean journey time.
A student was asked for the meaning of this confidence interval. The student replied as follows.
'The times for 98\% of these journeys are likely to be within the confidence interval.'
Explain briefly whether this statement is true or not.
Two independent 98% confidence intervals are found.
The widths，w cm ，of a random sample of 150 leaves of a certain kind were measured．The sample mean of w was found to be 3.12 cm ．
Using this sample，an approximate 95 % confidence interval for the population mean of the widths in centimetres was found to be［3．01，3．23］．
（a）Calculate an estimate of the population standard deviation．
（b）Explain whether it was necessary to use the Central Limit theorem in your answer to part（a）．［1］
Henri wants to choose a random sample from the 804 students at his college. He numbers the students from 1 to 804 and then uses random numbers generated by his calculator. The first 20 random digits produced by his calculator are as follows.
Henri's first two student numbers are 567 and 109.
Use Henri's digits to find the numbers of the next two students in the sample.
There were 30 students in Henri's sample. He asked each of them how much time, X hours, they spent on social media each week, on average. He summarised the results as follows.
Use this information to calculate an unbiased estimate of the mean of X and show that an unbiased estimate of the variance of X is less than 0.1 .
Henri's friend claims that Henri has probably made a mistake in his calculation of Σx or Σx2.
Use your answer to part (b) to comment on this claim.
A random sample of 250 people living in Barapet was chosen. It was found that 78 of these people owned a BETEC phone.
Calculate an approximate 98% confidence interval for the proportion of people living in Barapet who own a BETEC phone.
Manjit claims that more than 40% of the people living in Barapet own a BETEC phone.
Use your answer to part (a) to comment on this claim.
A random sample of 8 boxes of cereal from a certain supplier was taken. Each box was weighed and the masses in grams were as follows.
261249259252255256258254
Find unbiased estimates of the population mean and variance.
A student wishes to estimate the proportion, p, of students at her college who have exactly one brother. She surveys a random sample of 50 students at her college and finds that 18 of them have exactly one brother. She calculates an approximate α% confidence interval for p and finds that the lower limit of the confidence interval is 0.244 correct to 3 significant figures.
Find α correct to the nearest integer.
The numbers of green sweets in 200 randomly chosen packets of Frutos are summarised in the table.
| Number of green sweets | 0 | 1 | 2 | 3 | >3 |
|---|---|---|---|---|---|
| Number of packets | 32 | 50 | 97 | 21 | 0 |
Calculate an unbiased estimate for the population mean of the number of green sweets in a packet of Frutos, and show that an unbiased estimate of the population variance is 0.783 correct to 3 significant figures.
The independent random variables X and Y have the distributions Po(1.9) and Po(2.2) respectively．
A sample of 60 randomly chosen pairs of values of X and Y is taken, and the value of X+Y is calculated for each pair. The sample mean of these 60 values is found.
Find the probability that the sample mean of X+Y is less than 4.0 .