SCIE4402: Statistical analysis of different experiments and hypotheses - Statistics Assignment Help

Assignment Task

Assignment 1

Don’t start this assignment until you have completed and understood all exercises for Day 1 labs, as the questions here are very closely related to the relevant examples in the guided examples scripts covered in labs. When you have completed those exercises you should work through the following problems and note answer the questions as you go. When you’re confident take the Assignment 1 Quiz in LMS and enter the answers to these questions. You only have to enter answers into the LMS Quiz; you do not have to submit any written report. Remember that plagiarism is considered a serious offence at UWA – you are welcome to discuss and compare your work with other people but you must write your own script and do all the questions yourself. To show that you have done this you will have to upload the R script you use for the assignment when you submit your answers. Also note that some questions are specifically designed to check for plagiarism.

For any questions where you conduct an unpaired t-test, use the default version that does not assume equality of variance.

For numerical answers that are not whole numbers (eg. p-values), use at least 3 decimal places or significant figures of accuracy, as appropriate. Do NOT use scientific/exponential notation for numerical answers – the LMS will mark it as incorrect. An answer of 1.234e-5 should be entered as 0.00001234 or just 0 or 0.0001, since all are the same to three decimal places.

 

Problem 1.

A student plants 76 seedlings of a rare species to see if they will survive the summer. With 36 of the seedlings, she plants them with water-holding beads, while the rest are planted without. At the end of the summer, she finds that 27 of the seedlings she planted with water-holding beads have survived, while 21 of the ones planted without the beads have survived. Her hypothesis is that the beads should increase the survival of the seedlings.

1. a) What would be the best statistical test to use to test whether the student’s hypothesis is supported based on this data?

Two-proportion Z-test

1. b) Carry out the test. What is the p-value given by the test? Is her hypothesis supported?

H0: The survival of seedlings of both groups are equal.

H1: The survival of seedlings of both groups are not equal.

Since P-value = 0.2113 > 0.05 we failed to reject the null hypothesis that the survival of seedlings of both groups are equal.

c) What is the best estimate and the 95% confidence interval for the proportion of plants that will survive a similar summer in future if they are planted with water-holding beads?

The 95% confidence interval is between (-0.07545516, 0.40878850) for the proportion of plants to survive a similar summer in the future if they are planted with water-holding beads.

 

Problem 2.

Some researchers have developed a theory regarding the genetics determining resistance of a certain weed to a particular herbicide. According to this theory, one quarter of the plants raised in an experiment should be killed by the herbicide and three quarters should survive. When 50 plants are raised and sprayed with the herbicide, the researchers find that 32 of them survive and 18 die.

a) What would be the best statistical test to use to test whether the researchers’ hypothesis should be rejected based on this data?

Binomial test

1. b) Carry out the test. What is the p-value given by the test? Based on this sample, what is the 95% confidence interval for the proportion of plants that will survive the herbicide in future experiments with large numbers of plants? Should the researchers’ hypothesis be rejected?

H0: ¾ plants survived the herbicide.

H1: ¾ plants did not survive the herbicide.

Since, the p-value= 0.1004 > 0.05, we fail to reject the null hypothesis where ¾ plants survived the herbicide. The 95% confidence interval on the true proportion of survival is between (0.4919314 0.7708429).

c) What is the minimum number of plants surviving out of 50 that would have given a non-significant result in this case? And what is the minimum number of plants surviving out of 50 that would have caused the researchers’ hypothesis to be rejected because the number of plants surviving was too high?

The minimum number of plants surviving out of 50 plants that would have given a non-significant result is 32 (p-value= 0.1004, with 95% confidence interval between 0.4919314 0.7708429) and the minimum number of plants surviving out of 50 that could cause the researchers’ hypothesis to be rejected is 38.

 

Problem 3.

For this question we will use the hermit crab data that we used in the lab. We want to test whether there is a difference between hermit crabs fed on the fancy expensive pet shop mix ie ‘hermitcrabmix’ and crabs fed on home diets (home diets include both cornflakes and cornflakes plus vegies). We are interested in whether there are differences in terms of their weight or in terms of the amount they eat.

a) What would be the best statistical test to use?

T test

b) Test whether there is a difference in weights between the crabs fed on the pet shop diet vs the ones fed on a home diet. What is the p-value given by the test? Do we conclude that there is a significant difference?

H0: There is no significant difference in weights between the crabs fed with pet shop diet and the ones fed on a home diet.

H1: There is a significant difference between the crabs fed on the pet shop diet vs the ones fed on a home diet.

Since the p-value= 0.3952 > 0.05, we fail to reject the null hypothesis and conclude that there is no significant differences between the crabs fed with ‘hermitcrabmix’ and the ones fed with home diets.

c) Now test whether the crabs fed on the pet shop diet have a greater weight than the ones fed on a home diet. What is the p-value given by the test? Do we conclude that there is a significant difference?

H0: The crabs fed with ‘hermitcrabmix’ and home diets have no significant difference in weights

H1: The crabs fed with ‘hermitcrabmix’ has a greater weight compared to those fed with home diet.

Since the p-value= 0.1542>0.05, we fail to reject the null hypothesis and conclude that there is no significant proof that the crabs fed with ‘hermitcrabmix’ has a greater weight compared to the ones fed with home diet.

d) Now test whether the crabs fed on the pet shop diet eat a different amount than the ones fed on a home diet. What is the p-value given by the test? Do we conclude that there is a significant difference? How much, on average, do the crabs in the two groups eat?

Since the p-value= 0.6041> 0.05, we fail to reject the null hypothesis and conclude that there is no significant difference between the ones fed with ‘hermitcrabmix’ and the ones fed with home diet. The crabs consume between 0.52 and 0.47 grams respectively in the two groups.

 

Problem 4.

The data for this problem is in the ‘variety trial data.xlsx’ file available in LMS. An agronomist chooses 10 sites to trial two new varieties of wheat. At each site he chooses a single plot (same size at every site) and then divides it evenly into two. One side of each plot is randomly chosen for variety 1 and the other is used for variety 2. At the end of the season, each of the 20 subplots is harvested and the yield per hectare recorded in the ‘variety trial data.xlsx’ file.

The agronomist wants to know whether there is a real difference in yield between the varieties.

a) What would be the best statistical test to use?

Paired t-test

Carry out the test. What is the p-value given by the test? Can we conclude there a real difference in yield between the varieties? Based on the 95% confidence interval, what would we say is a reasonable lower bound on the difference between the yields of the two varieties? Which variety would we expect to yield more?

b) The next season a similar trial is conducted at a much larger number of sites. The data from this trial is found in the ‘variety trial data.xlsx’ file as well.

How many trial sites were used for the trial this time?

c) Sadly, there is an outbreak of a fungal disease across the trial region that badly affects some of the trial sites. The agronomist decides to only use any site where the mean yield for the site is more than 30% of the average yield for all the sites (ie sites that were less than 30% of average were scrapped). (Note, the mean yield for the site is the average of the yields for the two varieties at that site).

d) Create a subset of the data that fit the criteria. How many trial sites should be retained? (You may do this subsetting in R or in Excel as you prefer. If you can work it out, it will be much quicker if you use the R subset function like we did in labs! See the hints documents for assistance.)

e) Carry out the test for difference on this subset of sites. What is the p-value given by the test this time? Can we conclude there a real difference in yield between the varieties?

 

Problem 5.

Some herpetologists suspect that sex ratio in their favourite species of turtle may be affected by the temperature during incubation. They incubate 47 eggs at 26oC, and find that 27 of the baby turtles are females.

What is the best test to use, to test whether the observed sex ratio is significantly different to a 50-50 sex ratio?

What is the p-value obtained when you apply this test?

Would you conclude that the observed sex ratio is significantly different to a 50-50 sex ratio?

The herpetologists also incubate 87 eggs at 36oC, and find that 33 of the baby turtles are females.

What is the best test to use, to test whether the observed sex ratio is significantly different to a 50-50 sex ratio?

What is the p-value obtained when you apply this test?

Would you conclude that the observed sex ratio at 36oC is significantly different to a 50-50 sex ratio?

What is the best test to use, to test whether the observed sex ratio at 36oC is significantly different to the observed sex ratio at 26oC?

What is the p-value obtained when you apply this test?

Would you conclude that the observed sex ratios at the two different temperatures are significantly different?

Based on this evidence for this species, is incubation at higher or lower temperatures more likely to result in female baby turtles?

Some other herpetologists in a competing research team have another favourite species of turtle, which is related to the first species. This turtle has been studied quite a lot, so it is quite accepted that the male:female sex ratio of babies of this species incubated at 26oC is 7:3 (ie 70% of babies are male on average).

What is the best test to use, to test whether the observed sex ratio for the first species is significantly different to the accepted sex ratio for the second better-studied species?

What is the p-value obtained when you apply this test?

Would you conclude that the observed sex ratio for the first species is significantly different to the accepted sex ratio for the second better-studied species?

 

Problem 6.

Work through the ‘random numbers.R’ script trying to understand what’s happening with each line of code and answering the 19 questions (Q1, Q2, … Q19) in the script as you go.

Note that this problem is meant to be really challenging, especially towards the end. It tests your understanding of statistical concepts and R code at a much deeper level than the other problems. It’s up to you how much time and effort you want to put into it, but remember that you can get quite a good mark on the assignment without bothering with this problem at all! If you’re finding it stressful and/or very difficult to follow, then best to focus on the other questions and make sure you get the best possible marks for them. The rest of the course doesn’t depend on understanding this problem either.

 

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