1. A simple random sample of 101 cities and counties in the United States was selected and the number of positive COVID

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1. A simple random sample of 101 cities and counties in
the United States was selected and the number of positive COVID-19
cases per 10k citizens follows a normal distribution. The mean
number of positive cases for this sample of 101 counties and cities
was 253, with a standard deviation of 11.4. Assume that in Europe,
the mean number of positive COVID-19 cases per 10k citizens
is 𝜇0=μ0=298. Could we claim that the mean
number 𝜇μ of positive COVID-19 in the US is significantly
smaller than that 𝜇0μ0 in Europe?
(a) Write proper testing
hypotheses 𝐻0H0 and 𝐻1H1.
(b) Calculate the test statistic.
(c) Under the null hypothesis, which distribution does the test
statistic follow?
(d) Calculate the corresponding p-value.
(e) Draw your conclusion
2. David complains that the weather forecasts produced by
local radio are not better than those achieved by tossing a fair
coin and predicting rain if a head is obtained or no rain if a tail
is obtained. He records the weather for 50 randomly selected days.
The local radio forecast is correct on only 32 of these days. Could
we claim that the accuracy rate 𝑝 of the forecasts is
better than tossing a coin?
(a) Assume that the probability of correct forecasts by tossing
a coin is p0=0.5. State proper testing hypotheses.
(b) Calculate the test statistic.
(c) Calculate the corresponding p-value.
(d) Draw your conclusion