The following table contains summary statistics on house sales in Canberra (by region) for the last 3 months.

The following table contains summary statistics on house sales in Canberra (by region) for the last 3 months.
Region Mean House Price Number of houses sold
Belconnen $755481 210
Gungahlin $815826 228
Inner North $1180106 75
Inner south $1489622 42
Molonglo Valley $948222 28
Woden Valley $1065248 95
Weston Creek $802488 60
Tuggeranong $723422 228

The lake in the Centre of Canberra in Lake Burley Griffin. The first 3 regions in the table above (that is, Belconnen, Gungahlin and inner north) are north of the lake. The remaining regions are south of the lake.

Native Canberrans are friendly loyal to their side of the lake (that is, whether you live north or south lake burley). Those born and raised in north Canberra, tend to buy houses and retire in North Canberra, and would never consider moving south of the lake.

So how do Calculate the sample mean of house prices in North and South Canberra compare on house prices? You are going to use the summary statistics in the table above to assess whether there is sufficient evidence of a difference in the mean house price between North and South Canberra.

Note, for this question you may assume that:
⦁ The sample sizes in North and South Canberra are respectively large enough such that a t-test (of some sort) can be validly applied.
⦁ House prices are independent (that is, the sold price of one house is not affected by the sold price of any other house)
⦁ The houses that were sold in the last 3 months constitute a simple unbiased random sample of the houses within each region.

⦁ State the null (H0) and alternative (HA) hypotheses using proper statistical notation for the population parameters
⦁ Calculate the sample mean of house prices in North Canberra (that is, Belconnen, Gungahlin and inner north) (Please show your working by providing the equations you used to perform your calculation).
⦁ Calculate the sample mean of house prices in South Canberra (that is, Inner south, Molonglo Valley, Woden Valley, Weston Creek and Tuggeranong combined) (please show your working by providing the equations you used to perform your calculation).
⦁ The sample standard deviation of house prices in North and South Canberra are estimated to be $ 301250 and $476250 respectively. Will your test be a pooled-variance t-test or separate -variance t-test for the difference between two population means? Give a reason for your answer.
⦁ Calculate the value of your test statistic for this hypothesis test (Note: please show your working by providing the equation you used to calculate your test statistic)
⦁ Assuming a significance level of =0.05，what critical Value(s) define your rejection region?
(Note: if your degree of freedom is greater than 120, you may obtain your critical Values from the standard normal distribution

⦁ Do you have statistical evidence to support your alternative hypothesis given the data? Why or why not? State your conclusion in the context of the equation.
⦁ Calculate the p-value of your test (as extra value or interval range for the p-value may be given)
(Please also provide the mathematical expression you used to calculate your p-value)

⦁ What probability distribution would you assume the random variable X follow? Be specific and provide the values of the population parameters of your probability distribution or provide a formula for the probability distribution function P(X)
⦁ Using your answer to part (a), estimate the probability that more than 40% of the 200 smartphone users surveyed in Canberra have downloaded the COVIDSafe App.

⦁ State the null (H0) and alternative (HA) hypotheses of your test using proper statistical notation for the population parameters
⦁ Calculate the value of your statistic for this hypothesis test. (please show your working by providing the equation you used to calculate your test statistic)
⦁ Calculate the p-value of your test and interpret its meaning (Note: Please also provide the mathematical expression you used to calculate your p-value)

⦁ Do you have statistical evidence to support your alternative hypothesis given the data? (Assuming a significance level of =0.05) Why or why not? State your conclusion of the test in the context of the question.

⦁ Construct a 99% confidence interval estimate of the population proportion of smartphone users in Canberra who have downloaded the COVIDSafe APP. (Note: please show your working by providing the formula you used to calculate your confidence interval

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