Statistics Practice Questions And Solutions

Question 1(a)

Suppose that the amount of time teenagers spend on the Internet is normally distributed, with a standard deviation of 1.5 hours. A sample of 100 teenagers is selected at random, and the sample mean is computed as 6.5 hours.

• Determine the 99% confidence interval estimate of the population mean.
• Determine the 95% confidence interval estimate of the population mean, changing the sample size to 300.
• Comment on how the confidence interval and changes to sample size will impact on the interval estimates.

The amount of time spent by Australian adults playing sports per day is normally distributed, with a mean of 4 hours and standard deviation of 1.25 hours. Use this information to answer the following question(s).

• Find the probability that a randomly selected Australian adult plays sport for more than 5 hours per day.
• Find the probability that if four Australian adults are randomly selected, their average number of hours spent playing sport is more than 5 hours per day.
• Find the probability that if four Australian adults are randomly selected, all four play sport for more than 5 hours per day

At present, many universities in Australia are adopting the practice of having lecture recordings automatically available to students.  A university lecturer is trying to investigate whether having lecture recordings available to students has significantly decreased the proportion of students passing her course.  When lecture recordings were not provided to students, the proportion of students that passed her course was 80%.  The lecturer takes a random sample of 25 students, when lecture recordings are offered to students, and finds that 11 students have passed the course. Is there significant evidence to support this university lecturer’s claim? Use α = 0.01

A drug company is interested in the effectiveness of a new sleeping pill. A random sample of 50 people try the new sleeping pill and the number of additional hours of sleep (compared with the nights without any sleeping pill), X, are recorded. The sample mean of additional hours of sleep is 2.2 hours and the sample standard deviation of X is 3 hours.   

Test the claim that the new drug increases the number of hours of sleep at least by 2 hours on average at the 5% level of significance.

In 2003, computers of Brand A controlled 25% of the market, Brand B 20%, Brand C 10% and Brand D 45%. In 2004, sample data were collected from many randomly selected stores throughout the country. Of the 1200 computers sold, 280 were Brand A, 270 were Brand B, 90 were Brand C and 560 were Brand D. Has the market changed since 2003? Test at the 1% significance level

A major insurance firm interviewed a random sample of 1500 college students to find out the type of life insurance preferred, if any. The results follow:

	Insurance Preference
Gender	Term	Whole life	No insurance
Female	170	110	470
Male	195	75	480

Is there evidence that the life insurance preference of male students is different to that of female students? Test using the 5% level of significance.

Question 3(c)

The following data are believed to have come from a normal probability distribution.

26	21	25	20	21	29	26	23	22	24
24	30	23	32	26	24	32	16	36	26
21	31	26	23	32	35	40	30	14	26
46	27	33	25	27	21	26	18	29	36

The mean of this sample equals 26.80, and the standard deviation equals 6.378.

Investigate at the 5% significance level to test this claim.

Pop-up coffee vendors have been popular in most campuses world-wide. A potential vendor is interested in exploring the possibility of operating a pop-coffee in Darwin. People she spoke to are trying to discourage her as they claim that the median temperature of 38 degrees Celsius is too hot for people to consume coffee through an open transportable vending facility. You have been appointed to offer assistance in knowing how temperature (in degrees Celsius) impacts daily hot coffee sales revenue (in $00’s).

The potential vendor has collected using a random sample for 6 days to generate a link between the daily hot coffee sales revenue and the corresponding temperature of that day noted.  

Coffee sales revenue	Temperature
6.50	25
10.00	17
5.50	30
4.50	35
3.50	40
28.00	9

An Excel regression output was obtained on this data which is given below.

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.8644
R Square	0.7472
Adjusted R Square	0.6840
Standard Error	5.2027
Observations	6
ANOVA
	df	SS	MS	F	Significance F
Regression	1	320.0617	320.0617	11.8244	0.0263
Residual	4	108.2716	27.0679
Total	5	428.3333
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	27.7179	5.6629	4.8946	0.0081	11.9952	43.4406
Temperature	-0.6943	0.2019	-3.4387	0.0263	-1.2549	-0.1337

(a) Estimate daily hot coffee sales revenue on a day of 38 degrees Celsius.

(b) State with reasons why your prediction in part (a) is reasonable or not?

(c) Test the significance of the slope, against a two-tailed alternative, at the 5% level of significance.   

(d) Prepare a short report to the potential vendor of your analysis including your recommended decisions.

A statistics course at a large university is taught in each semester.  A student has noticed that the students in semester 1 and semester 2 are enrolled in different degrees. The student believes that the cohort of students in semester 1 do better than in Semester 2. To investigate, the student takes a random sample of 25 students from semester 1 and 25 students from semester 2 and records their final marks (%) provided in the table below.  Excel was used to generate descriptive statistics on each sample. Based on past experience we know that final marks in statistics course have a normal distribution.  

Sample of semester 1 final marks					Sample of semester 2 final marks
65	45	53	76	53	45	40	53	58	75
85	55	63	85	77	46	82	54	75	59
45	57	55	60	83	45	54	87	77	63
56	83	55	52	67	81	60	56	53	65
82	64	62	88	71	52	65	60	65	54

Semester 1		Semester 2
Mean	65.48	Mean	60.96
Standard Error	2.679	Standard Error	2.5136
Median	63	Median	59
Mode	55	Mode	54
Standard Deviation	13.395	Standard Deviation	12.568
Sample Variance	179.43	Sample Variance	157.96
Range	43	Range	47
Minimum	45	Minimum	40
Maximum	88	Maximum	87
Sum	1637	Sum	1524
Count	25	Count	25

Comment on the two semester grades detailing the steps you would take (based on session 1 to 11 discussions in this unit) and investigate if student’s in different semesters do produce different grades.

Before you begin, please enter your personal details in the table below. These will be used to calculate your recommended number of serves for each food group and Recommended Daily Intakes (RDIs) or Adequate Intakes (AIs) for your three chosen nutrients.

Name	Candice
Age	19
Gender	Female

Name: Candice Date: 30th September Weekday or weekend: Saturday

Time	What I ate and drank	Amount (cups, mls or grams)	Vegetables	Fruit	Grain (cereal) foods	Lean meat and alternatives	Dairy and alternatives	Discretionary foods
9:30 am	Bran flakes	2/3		Apricot			Milk
	1 Apple	1		Apple
1:30 pm	Sandwich	2			Whole wheat grain		Margarine
	Asparagus salad	1 (75gm)	Asparagus, bell pepper		Pecan nuts
	Roasted Beef	60gm				Beef
5:00 pm	Muffin	1 (35gm)	Carrot	Orange juice	All purpose flour		Butter	Carrot muffin
10:00pm	Pasta	½ (80gm)	Onion, tomatoes			Egg, red kidney beans
	Salad	¼ (75gm)	Lettuce, cucumber			Peanuts	Cheddar
TOTAL SERVES			4	2	4	2½	3	2 ½

Name: Candice Date: 2^nd October Weekday or weekend: Monday

Time	What I ate and drank	Amount (cups, mls or grams)	Vegetables	Fruit	Grain (cereal) foods	Lean meat and alternatives	Dairy and alternatives	Discretionary foods
8am	Spring roll [with pork and cabbage inside]	4 (40gm)	cabbage			Pork
	2 eggs [made it into soup]	2 (120gm)			eggs
12pm	Beef stir-fried with tomato	1 (65gm)	tomato			Beef
	Rice	1 (100gm)			rice
	2 scoops vanilla ice cream	75gm						Vanilla icecream
7pm	Bouillon	1 (75gm)	Potato, carrot, celery, tomato
	2 slices whole wheat toast	2 (40gm)			Wheat grain
9:30 pm	Chicken breast	80gm				Chicken
	Broccoli salad	1 (75gm)	Broccoli, Red onions				Cheddar, mayonnaise
TOTAL SERVES			4		5	3	1	2 ½

Enter the number of serves of each food group you ate each day into the table below and calculate the average. Use the recommended food serves calculator at to work out the number of serves of each food group you should be eating each day and enter into the table below your average.

Time	Vegetables	Fruit	Grain (cereal) foods	Lean meat and alternatives	Dairy and alternatives	Discretionary foods
DAY 1	4	2	4	2 ½	3	2 ½
DAY 2	4	0	5	3	1	2 ½
AVERAGE NUMBER OF SERVES	4	1	4 ½	3 ¾	2	2 ½
RECOMMENDED NUMBER OF SERVES*	5	2	6	2 ½	2 ½	0-2 ½

Enter the recommended daily intakes (or adequate intakes) for allof the six nutrients in the following table. You can find the values at by entering your personal details.

Then, using NUTTAB as shown in the tutorial in week 4, calculate the average amount of the three nutrients you chose to analyse from the foods that you

	RECOMMENDED DAILY INTAKE (RDI) (OR ADEQUATE INTAKE (AI))	YOUR AVERAGE INTAKE
PROTEIN (g/day)	46
FIBRE (g/day)	25	30.07
SODIUM (salt) (mg/day)	460-920
CALCIUM (mg/day)	1000	452.85
IRON (mg/day)	18	18.3155
FLUIDS (mL/day)	2100

Using the information you have calculated on the previous pages, answer the following questions. Type your answer straight into this word document, under each question. Be sure to stick to the word limit given for each question. This report section will be a large proportion of your marks for the assignment. Refer to the marking criteria for more information.

We would recommend that instead of rewriting the figures from the nutrient tables, that you refer to them in-text eg Table 3 shows XYZ. This will help keep your word count down. Read each question carefully and do not skip any.

1. Briefly discuss how your eating patterns tie into the concept of ‘holistic health’ and your philosophy of eating. Is your food diary a typical representation of your daily diet?  If not, why not?

My eating habits fall into holistic health pattern. I intend to eat healthy food items 80% of the time, and indulge in treat food for 20%. It helps me to stick to a healthy diet plan, avoid unnecessary weight gain and remain healthy. Following a strict diet chart is not my intention. I consume treat meals not more than twice a week. However, sometimes I deviate from my healthy eating policy when it comes to holidays due to unavailability of home cooked food. The food diary accurately depicts my eating habits. Furthermore, my weekend diary deviates a bit from a typical day as I wake up late, indulge in discretionary food for evening snack and usually have late dinner. It was overall a learning experience of keeping a track of my food consumption throughout 2 days. It helped me evaluate how good or poor my eating habit had been for that day.

2. Discuss and link how well your average intake of the five food groups and discretionary foods (seen in Table 2, calculated from your food diary) follow the Australian Dietary Guidelines recommendations across the two days recorded. Are any food groups significantly high or low in intake? If so, is this because you make a conscious effort to eat/avoid these foods? You may round your food groups up or down to the nearest half so comparison is easier.

Question 1(b)

The Australian Dietary Guidelines recommend consumption of adequate nutritious drinks and food products to meet the energy requirements of the body. My average intake is slightly below the daily recommended intakes of all food groups. My food intake on weekends is more towards the recommended intake requirements. The consumption of fruits is negligible on weekdays. I have always found consuming whole fruits or juices to be very tiresome and unappealing. This area requires improvement. My food diary abides by the guideline of having a wide variety of nutritious foods belonging to the different groups. I eat plenty of vegetables, lean meat and cereal foods. Moreover, I drink plenty of water and abstain myself from alcohol consumption. However, I need to restrict my habit of adding salts to foods at the table. This may affect my health and needs to be limited. I also need to include low fat dairy products in my eating habits. This will increase the amount of calcium, vitamin A and D in my system. Inclusion of dairy products and fruits will help to prevent the occurrence of cardiovascular disease and several forms of cancer. The guidelines recommend limiting intake of fruits that contain predominantly high level of saturated fats such as cakes, pastries, biscuits, processed meat or fried snacks. I always try to follow this guideline when it comes to healthy eating. I indulge in treat meals like muffins or ice-creams occasionally. On most of the days, I try to incorporate maximum amount of cereals, high fiber containing food products and vegetables in my diet. My food diary shows a comparatively low intake of legumes and pulses. My intake of legumes is restricted to kidney beans and some nuts. Therefore, I need to include more of this food group into my food habit.

3. State which three nutrients you chose to analyse and after referring the reader back to Table 3, were there any nutrients that were significantly higher or lower than the recommended intake appropriate for your age and gender? For your three chosen nutrients, critically analyse your foods and beverages and link the medium to high intakes of the nutrients to which foods would have contributed towards these levels. If you consumed less than 50% of the RDI/AI for any of the three nutrients, which foods or meals are rich sources of that nutrient that you should incorporate into your diet?

The three nutrients I chose to analyze from the nutrient requirement table are calcium, iron and dietary fiber. The daily nutrient requirements as per the dietary pattern of a female belonging to the age group of 19-20 years are 1000 mg/day for calcium, 18 mg/day for iron and 25g/day for dietary fiber. My average consumption chart shows that the intake levels for all the above mentioned three nutrients are below the recommended levels. However, the amount of fiber and iron that I receive regularly are below the recommended amounts by minute limits. The average daily intake of fiber is 30.07 g/day, which is more than the daily body requirement. The average iron intake is 18.3155 mg/day. It matches the amount that is needed by our body. Thus, the foods that I incorporate in my regular diet are adequate to meet the necessary nutrient requirements of iron and dietary fiber. My average nutrient intake shows that the nutritional needs for calcium are not met by my body. The average weight of an adult is made up of around 2% calcium. Calcium gets incorporated in your body principally through consumption of yoghurt, milk and cheese. Packaged foods that are fortified with calcium and green leafy vegetables also help to increase calcium intake in the body to a certain extent. The huge deficit of calcium in my diet is making me vulnerable to diseases such as osteoporosis. Inadequate consumption of dairy products leads to addition of calcium to the body. I do not consume dairy foods. This acts as a major contributing factor in the low calcium levels in my body (Guenther et al., 2013).

Question 2(a)

Moreover, calcium is needed for girls belonging to my age group to build peak mass of the bones and maintain them. My daily calcium intake of 452.85 mg/day is less than 50% of the RDI. Regular consumption of bran flakes, whole wheat bread, salad and beef is responsible for providing my body with the necessary amount of iron. I had included apricots, red meat, poultry, pork, breads and vegetable salad in my diet. These food items are rich in iron content (Sizer et al., 2012). Consumption of meat helped my body to absorb more iron when compared to other food sources. My eating habit meets the daily required amount of fiber due to the presence of bran flakes, apple, salad and whole wheat bread. My high fiber diet will reduce the risks of heart disease, hypertension and stroke. The intake of fiber rich food products help in lowering my blood glucose and cholesterol levels (Schwingshackl & Hoffmann, 2015). If I ever skip intake of such food, that would decrease the amount of fiber in my body and may lead to constipation. Moreover, the adequate levels of fiber speed up the process of digestion and create a sense of satiety. In order to meet the amount of calcium in my diet, I need to have at least a glass of milk with protein powder regularly.

Consumption of 300 mg yoghurt or kefir, cheese and sardine fish will fulfill my daily calcium needs. I should consume a meal of sardine once a week to maintain the calcium intake. Parmesan or Mozzarella cheese will act as a good source of calcium. Figs will also provide adequate amounts of calcium and iron (Friel, Barosh & Lawrence, 2014). Moreover, its high fiber content will also prove beneficial. Apart from dairy products, consumption of sea food like oysters and prawn will help me gain benefits of extra calcium. Consumption of tofu, which is made from soy milk on alternate days and regular consumption of a handful of almond nuts will provide a boost of calcium and help me to meet the RDI.

4. Using appropriate literature discuss any potential implications of your current intake. For example any diseases that you are at risk of developing if your intake remains as is. If by chance your intake is perfect, discuss the positive influence of your diet on your future health.

A study was conducted by Hunnicutt & Xun, (2014) to investigate the relationship between dietary iron intake and incidence of coronary heart disease by conducting a metaanalysis of cohort studies. Peer reviewed journals were searched by using PubMed and Embase. Fixed or random effects model were used to analyse the incidence of CHD among people who consumed greater amounts of dietary iron. On analysis, it was found that CHD incidence was positively associated with the amount of heme iron in the diet (RR: 1.57; 95% CI: 1.28, 1.94). However, the study failed to establish any significant association between the iron intake and mortality.  Another study aimed to examine the effects of iron intake among vegetarian children. On measuring dietary iron intake, blood count and iron rich food sources, it was found that children who were vegetarian and omnivores, failed to meet the adequate iron levels that are required by their body. Prevalence of iron deficiency was higher among vegetarian in spite of consumption of Vitamin C rich food (Gorczyca et al., 2013).

Question 2(b)

The association between dietary iron intake and type 2 diabetes incidence was investigated among post-menopausal women. Larger dietary iron intake was found to be linked with increased risk of type 2 diabetes among alcoholic women (95% CI, 1.29–7.12) (Bao et al., 2012). A study examined the association of long term dietary fiber intake and the incidence of ulcerative colitis (UC) and Chron’s disease (CD). On analyzing 170,776 women and performing a follow-up over 26 years, a 40% reduction in CD was observed associated with increased intake of fruits, whole grains, cereals and legumes. However, the association with low UC incidence rate was not established (Ananthakrishnan et al., 2013).  Another study determined whether greater intake of fiber was associated with a decrease in mortality and inflammation related to chronic renal disease. It also aimed to assess the influence of renal disease on reduction of inflammation and its association with fiber intake.

Data analysis from 14,543 participants revealed that serum C-reactive protein levels decreased by 38% in people with renal disease, who consumed 10-g/day fiber.  An inverse relation was established between dietary fiber intake mortality among patients suffering from renal disease (Krishnamurthy et al., 2012). Dietary fiber health benefits were determined by another study conducted by Kaczmarczyk et al., (2012).Upon research, it was found that dietary fiber prevented the incidence of colon cancer, cardiovascular disease and type 2 diabetes mellitus. These fibers modulate ingestion, digestion and absorption of food particles, thereby reducing the risk of hypercholesterolemia, hyperlipidemia and hyperglycemia. A cohort data analysis showed that dietary calcium intake led to a statistically significant reduction in myocardial infarction (Li et al., 2012). On the other hand, high intake of calcium supplements was shown to be associated with excess risk of cardiovascular disease among men (RR_{>1000 vs 0 mg/d}, 1.20; 95% CI, 1.05-1.36) (Xiao et al., 2013). An increase in dietary calcium intake and consumption of milk products showed positive effects in maintaining bone health and preventing osteoporosis (Hong, Kim & Lee, 2013). Thus, I can conclude from the literature search that my diet makes me more vulnerable to osteoporosis, myocardial infarction and other cardiovascular disorders.

5. What are the main improvement that need to be made to better meet your recommendations? Please discuss five practical changes you could implement in your daily diet to make these improvements and be specific with which foods this could include and how you would make substitutions to your current diet successfully.

In order to improve my diet to meet the recommended amount of nutrients, I need to implement certain steps:

• I should keep a track of my daily food intake and should use a calorie counter to check whether I am consuming more energy than the amount I am expending. I should start enjoying a wide variety of foods that are rich in vitamins (carrot and pumpkin for vitamin A; citrus fruits for vitamin C and fatty fish and beef liver for vitamin D). I should add more lean meat varieties in my eating habit and cook them without using any fat (use grill or barbecue). I should remove the skin of the meat and trim the fat properly from before cooking it.
• I should not avoid consumption of starchy food to lose weight. Starchy food like brown-rice, whole-wheat bread and potatoes should be included in my diet as they contain less salt and more fiber. In order to maintain my daily iron requirements, I would have to consume oranges, berries, capsicum, kiwi and other fruits that will act as iron supplements.
• Daily intake of a glass of orange juice will also assist my physiological system to absorb dietary iron in a better way (Abbaspour, Hurrell & Kelishadi, 2014). I should add fruits as a topping to sandwich toast and breakfast cereal. Adding new cereals in my diet such as polenta or couscous would add bulk to the diet without increasing fat content (Hingle et al., 2014).
• On the other hand, if I increase the consumption of fiber rich foods like, artichokes, lima beans, Brussels sprouts, peas, avocados and oat meals, the food will pass through the intestine quickly and will lead to less absorption of minerals. That can create uncomfortable feelings such as cramping and bloating. Therefore, I need to maintain my intake of fiber rich food and no overdo it. If I choose not to include meat in my diet, I will have to increase the intake of iron-rich vegetables like mushrooms, lemongrass, chickpeas, black beans, parsley, lentils and spinach.
• I should snack on 5 servings of green leafy vegetables and at least 2 fruits instead of cakes, chips and other junk food. Every meal should be accompanied with dairy product to help me fulfill the calcium requirements. I would practice drinking plenty of water and avoid consumption of alcohol, sweets and coffee. Avoiding carbonated beverages like soda and cola would help in reducing calorie intake.

References

Abbaspour, N., Hurrell, R., & Kelishadi, R. (2014). Review on iron and its importance for human health. Journal of research in medical sciences: the official journal of Isfahan University of Medical Sciences, 19(2), 164.

Ananthakrishnan, A. N., Khalili, H., Konijeti, G. G., Higuchi, L. M., De Silva, P., Korzenik, J. R., ... & Chan, A. T. (2013). A prospective study of long-term intake of dietary fiber and risk of Crohn's disease and ulcerative colitis. Gastroenterology, 145(5), 970-977.

Bao, W., Rong, Y., Rong, S., & Liu, L. (2012). Dietary iron intake, body iron stores, and the risk of type 2 diabetes: a systematic review and meta-analysis. BMC medicine, 10(1), 119.

Friel, S., Barosh, L. J., & Lawrence, M. (2014). Towards healthy and sustainable food consumption: an Australian case study. Public health nutrition, 17(5), 1156-1166.

Gorczyca, D., Prescha, A., Szeremeta, K., & Jankowski, A. (2013). Iron status and dietary iron intake of vegetarian children from Poland. Annals of Nutrition and Metabolism, 62(4), 291-297.

Guenther, P. M., Casavale, K. O., Reedy, J., Kirkpatrick, S. I., Hiza, H. A., Kuczynski, K. J., ... & Krebs-Smith, S. M. (2013). Update of the healthy eating index: HEI-2010. Journal of the Academy of Nutrition and Dietetics, 113(4), 569-580.

Hingle, M. D., Wertheim, B. C., Tindle, H. A., Tinker, L., Seguin, R. A., Rosal, M. C., & Thomson, C. A. (2014). Optimism and diet quality in the Women's Health Initiative. Journal of the Academy of Nutrition and Dietetics, 114(7), 1036-1045.

Hong, H., Kim, E. K., & Lee, J. S. (2013). Effects of calcium intake, milk and dairy product intake, and blood vitamin D level on osteoporosis risk in Korean adults: analysis of the 2008 and 2009 Korea National Health and Nutrition Examination Survey. Nutrition research and practice, 7(5), 409-417.

Hunnicutt, J., He, K., & Xun, P. (2014). Dietary iron intake and body iron stores are associated with risk of coronary heart disease in a meta-analysis of prospective cohort studies. The Journal of nutrition, 144(3), 359-366.

Kaczmarczyk, M. M., Miller, M. J., & Freund, G. G. (2012). The health benefits of dietary fiber: beyond the usual suspects of type 2 diabetes mellitus, cardiovascular disease and colon cancer. Metabolism, 61(8), 1058-1066.

Krishnamurthy, V. M. R., Wei, G., Baird, B. C., Murtaugh, M., Chonchol, M. B., Raphael, K. L., ... & Beddhu, S. (2012). High dietary fiber intake is associated with decreased inflammation and all-cause mortality in patients with chronic kidney disease. Kidney international, 81(3), 300-306.

Li, K., Kaaks, R., Linseisen, J., & Rohrmann, S. (2012). Associations of dietary calcium intake and calcium supplementation with myocardial infarction and stroke risk and overall cardiovascular mortality in the Heidelberg cohort of the European Prospective Investigation into Cancer and Nutrition study (EPIC-Heidelberg). Heart, 98(12), 920-925.